Constructions of Complex Equiangular Lines
by
Amy Wiebe
B.Sc., Simon Fraser University, 2010
a Thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science
in the
Department of Mathematics
Faculty of Science
c Amy Wiebe 2013
SIMON FRASER UNIVERSITY
Fall 2013
All rights reserved.
However, in accordance with the Copyright Act of Canada, this work may be
reproduced without authorization under the conditions for “Fair Dealing.”
Therefore, limited reproduction of this work for the purposes of private study,
research, criticism, review and news reporting is likely to be in accordance
with the law, particularly if cited appropriately.
APPROVAL
Name:
Amy Wiebe
Degree:
Master of Science
Title of Thesis:
Constructions of Complex Equiangular Lines
Examining Committee:
Dr. Marni Mishna
Associate Professor of Mathematics
Chair
Dr. Jonathan Jedwab
Professor of Mathematics
Senior Supervisor
Dr. Matthew DeVos
Associate Professor of Mathematics
Supervisor
Dr. Karen Yeats
Assistant Professor of Mathematics
SFU Examiner
Date Approved:
December 3rd, 2013
ii
Partial Copyright Licence
iii
Abstract
A set of unit vectors in Cd represents equiangular lines if the magnitudes of the inner product
of every pair of distinct vectors in the set are equal. The maximum size of such a set is d2 ,
and it is conjectured that sets of this maximum size exist in Cd for every d ≥ 2. In this thesis
we describe evidence supporting the conjecture. We explain the fiducial vector method of
construction, outlining the methods of approach and combining the viewpoints of several
authors responsible for known examples of maximum-sized sets of equiangular lines. We
also identify several milestone publications and note specific examples of maximum-sized
sets of equiangular lines which we believe to be key in eventually resolving the conjecture.
Furthermore we give two new maximum-sized sets of equiangular lines in dimension 8; one
set is simpler than all previously known examples and the other illuminates previously
unrecognized underlying structure through connections with other dimensions. Using these
sets of lines we are able to demonstrate some important connections between equiangular
lines and other combinatorial objects, including Hadamard matrices, mutually unbiased
bases and relative difference sets.
iv
Acknowledgments
I would like to thank ...
My supervisor, Jonathan Jedwab, for his guidance throughout this entire process and
for believing that I had the ability to tackle such an immense problem. Thank you for
being the first person to introduce me to mathematics research and for making me want
to pursue graduate studies in the first place.
Jim Davis for taking time to delve into this problem, immerse himself in the work I had
done and provide suggestions which helped to get me unstuck at a critical time in the
research process.
Matt DeVos for his insightful questions and comments on this thesis, and his seemingly
innate ability to turn conjectures into theorems.
Simon Fraser University and NSERC for their generous funding which allowed me to
focus on my research.
My family and friends for their unconditional support and many attempts to understand
exactly what it is that I do.
v
Contents
Approval
ii
Partial Copyright License
iii
Abstract
iv
Acknowledgments
v
Contents
vi
1 Overview
1
2 Background and Previous Work
9
2.1
Equiangular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Mutually Unbiased Bases and Difference Sets . . . . . . . . . . . . . . . . . . 15
2.3
A Simple Example of 16 Equiangular Lines for d = 4 . . . . . . . . . . . . . . 27
2.4
Hoggar’s 64 Equiangular Lines for d = 8 . . . . . . . . . . . . . . . . . . . . . 28
2.5
Zauner’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5.1
Group-Covariance (A First Refinement of Zauner’s Conjecture) . . . . 29
2.5.2
Fiducial Vector Construction (A Second Refinement of Zauner’s Conjecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6
Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7
Clifford Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 New Constructions
52
3.1
Construction from MUBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2
Construction from a Complex Hadamard Matrix . . . . . . . . . . . . . . . . 63
vi
3.3
Construction from Building Blocks in a Smaller Dimension . . . . . . . . . . 73
4 Conclusions and Questions
82
Bibliography
84
Appendix A
87
A.1 d = 8 Fiducial Vectors [3, 39] . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A.2 d = 9 Fiducial Vector [4, 39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
A.3 d = 12 Fiducial Vector [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
A.4 d = 16 Fiducial Vector [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Index
102
vii
Chapter 1
Overview
Equiangular lines were first studied in 1948 by Haantjes in [29]. He studied equilateral ptuples in elliptic space, which are a set of p points such that all pairs of points are separated
by the same distance. The primary interest in this problem is finding the maximum size
p for which a set of p equilateral points exists in elliptic d-space. Haantjes [29] found
this maximum to be 6 for both 2-space and 3-space. Van Lint and Seidel [40] then found
the maximum to be 10 for 4-space, 16 for 5-space and at least 28 for 6-space. Using the
projective model of elliptic geometry, this problem in elliptic d-space can be recast as a
problem in (d + 1)-dimensional real space, where points become lines through the origin
and the distance between points becomes the angle between lines. The study of equiangular
lines in real space is connected to graph theory, where such lines can be viewed as regular
two-graphs [18, 24]. Their study also precipitated the introduction of what is now commonly
known in graph theory as the Seidel adjacency matrix of a graph [40].
In Rd the absolute upper bound (depending only on the dimension d, not on a given
value for the common angle between lines) on the size of a set of equiangular lines, attributed
to Gerzon, is given in [37, Theorem 3.5] as d+1
2 . However, it can be seen that the actual
maximum number of equiangular lines attained in Rd often falls short of this bound [18, 37].
Equiangular lines are also studied in complex space, Cd . In Cd the absolute upper bound
on the size of a set of equiangular lines is d2 . We will see that having inner products of
constant magnitude for all distinct pairs of vectors in a set in which all vectors have equal
norm is equivalent to having a set of equiangular lines.
1
2
CHAPTER 1. OVERVIEW
Example 1. In C2 a maximum-sized set of equiangular lines is given by the 4 vectors
√
√
( 12 ( 2 + 6)
√
√
( 12 ( 2 + 6)
(
1
(
1
1
)T
)T
√
1
T
2 ( 2 + 6)i )
√
√
− 12 ( 2 + 6)i )T
√
−1
Notice that the magnitude of the inner product between any distinct two of the given vectors
√
√
√
√
√
is | 41 ( 2 + 6)2 − 1| or | 12 ( 2 + 6)(1 ± i)|, which is equal to 1 + 3.
In quantum theory, a set of equiangular lines is known as a symmetric informationallycomplete positive operator-valued measure (SIC-POVM). POVMs represent measurements
in quantum theory. SIC-POVMs are particularly desirable measurements as their statistics completely determine the quantum state on which the measurement is carried out
(informationally-complete), and all pairwise inner products of POVM elements are equal
(symmetric) [38].
SIC-POVMs are a candidate for “standard quantum measurement”
[20, 21, 38], and additionally have applications to quantum tomography [15], quantum
cryptography [21], Kochen-Specker arguments [12], high precision radar [33] and speech
recognition [7]; they are also important foundationally to the “QBist” approach to interpreting quantum mechanics [5]. Mathematically, they find connections to spherical codes
[19, 38], mutually unbiased bases [2, 11, 23, 25], the theory of Lie Algebras [6] and elliptic
curves [11].
In contrast to the real case, it was conjectured by Zauner [42] that the absolute upper
bound for the complex case is attainable in all finite dimensions.
Conjecture 2. For each integer d ≥ 2, there is a set of d2 equiangular lines in Cd .
Equiangular lines have been constructed analytically (exact solutions) only in dimensions
2–16, 19, 24, 28, 35, 48, though sets of lines which are equiangular up to machine precision
(numerical solutions) exist in all dimensions d ≤ 67.
In 1975, Delsarte, Goethels and Seidel [18] tackled dimensions 2 and 3, stating that
maximum-sized sets of equiangular lines contain 4 and 9 lines, respectively, but leaving the
details of constructing such sets to [16]. In 1981, Hoggar [30] gave a set of equiangular lines
in C8 , which originated from the construction of polytopes in quaternionic 4-space. Hoggar
originally gave 64 lines as vectors over the quaternions; however, upon consideration of the
complexified versions of these vectors, he also showed that they correspond to 64 equiangular
CHAPTER 1. OVERVIEW
3
lines in complex 8-space [32]. In 1982, Hoggar [31] gave many examples of sets of lines in
various spaces whose angle sets are small. Included in these examples are maximum-sized
sets of equiangular lines in dimensions 2 and 3, and a restatement of his previous example in
dimension 8, all of which he discovered significantly before the study of complex equiangular
lines was popularized. More extensive study of equiangular lines in complex space has only
occurred recently with the emergence of the importance of equiangular lines to quantum
information theory.
Complex equiangular lines were considered in more detail by Zauner in his 1999 thesis
[42], under the description regular quantum designs with degree 1, r = 1 and d2 elements.
In this thesis, he gave new exact solutions for d = 3, as well as exact solutions for d = 4, 5
and numerical solutions for d = 6, 7. With this evidence he proposed Conjecture 2, that
there exist exact solutions for all finite dimensions. Zauner also introduced what is currently
the standard method of construction of equiangular lines. This construction builds a set
of equiangular lines as the orbit of an initial vector under the action of a matrix group.
The initial vector is called a fiducial vector. The matrix group used by Zauner is the WeylHeisenberg group of matrices and his fiducial vector is a special eigenvector of a particular
matrix in the Clifford group. Zauner further conjectured that it is always possible to generate
equiangular lines in this way.
Conjecture 3. For all d ≥ 2, there exists a fiducial vector which is an eigenvector of a
particular matrix in the Clifford group, such that a set of d2 equiangular lines is generated
by the action of the Weyl-Heisenberg group on the fiducial vector.
More widespread interest in the problem was generated by a 2004 paper by Renes et al.
[38]. While the authors did not find any new exact solutions, they investigated connections
between SIC-POVMs and several other areas of study. Using connections to frame theory,
they found fiducial vectors numerically for d ≤ 45 by solving an optimization problem;
they showed that SIC-POVMs correspond to spherical 2-designs; and that four groups, in
addition to the Weyl-Heisenberg group, induce numerical SIC-POVMs (in particular, in
dimensions 6, 8 and 9). However, they did not address Conjecture 3, that fiducial vectors
always occur as eigenvectors of a particular matrix.
The numerical solutions in [38] were in fact shown to satisfy the conditions of Conjecture 3 by Appleby in [1]. In this same paper, Appleby constructed exact fiducial vectors
for d = 7, 19. Based on the simplification he made in these dimensions, he suggested that
CHAPTER 1. OVERVIEW
4
there might be an infinite sequence of dimensions for which finding fiducial vectors simplifies. However, Khatirinejad [35] showed that the method of construction used by Appleby
does not directly generalize to higher dimensions. Khatirinejad also gave alternative exact
solutions for d = 7, 19 using the Legendre symbol, but showed that this method also does
not generalize to higher dimensions.
The first non-prime-power dimension for which an exact solution was found was d = 6,
by Grassl [25]. This solution was found via symbolic calculation on a computer algebra
system and was shown to satisfy the conditions of Conjecture 3. In [39], Scott and Grassl
found fiducial vectors numerically for d ≤ 67 by minimizing an equation obtained via a
characterization of SIC-POVMs in terms of t-designs. They also found exact solutions for
more non-prime-power dimensions, d = 14, 24, 35, 48, by analyzing the numerical solutions
they obtained.
In order to find exact solutions, assumptions on the structure of the set of equiangular
lines are often made in order to restrict the search space of fiducial vectors. These assumptions generally include some symmetry condition with respect to the group which acts on
the fiducial vector. In particular, the set of lines is usually assumed to be group covariant,
which means that the set of lines is closed under the action of the group, and that the group
acts transitively on the set of lines. Most known examples of equiangular lines are group
covariant with respect to the Weyl-Heisenberg group. However, even with such assumptions, the search for fiducial vectors is far from trivial. In [4], the authors described a new
basis in which to view the Weyl-Heisenberg group for square dimensions which simplified
calculations for previously solved dimensions d = 4, 9 and enabled the calculation of an
exact solution for d = 16. Furthermore, in [3] the authors extended this result, discussing
systems of imprimitivity of the Clifford group which they used to reconstruct exact solutions
in dimensions d = 8, 12 and to construct a new exact solution for d = 28.
Table 1.1 summarizes the dimensions for which exact and numerical solutions are known,
along with the author(s) to whom their existence is due.
In this thesis we first describe the traditional approach to finding sets of d2 equiangular
lines in Cd . In Chapter 2, we describe previous research on equiangular lines. In §2.1 and
§2.2, we introduce the major objects studied in this thesis. In §2.3 and §2.4 we make note
of two maximum-sized sets of equiangular lines that we believe to be suggestive of a new
approach to the problem. In §2.5–2.7 we detail the standard fiducial vector method of
construction of maximum-sized sets of equiangular lines. We state several conjectures that
5
CHAPTER 1. OVERVIEW
Table 1.1: Known Maximum-sized Sets of Equiangular Lines
Source
Delsarte, Goethals & Seidel [18]
Hoggar [30, 32]
Year(s)
Dimensions
Type
1975
2,3
exact
1981, 1982
8
exact
2,3,4,5
exact
Zauner [42]
1999
Renes et al. [38]
2004
Appleby [1]
6,7
numerical
2,3,4
exact
≤ 45
numerical
2005
7,19
exact
2004–6, 2008
6,8-13,15
exact
Bos & Waldron [14]
2007
2,3,5,7
exact
Khatirinejad [35]
2008
7,19
exact
Grassl & Scott [39]
2010
14,24,35,48
exact
Appleby et al. [4]
Appleby et al. [3]
Grassl [25, 26, 27, 28]
≤ 67
numerical
2011
4,9,16
exact
2012
8,12,28
exact
have been made regarding this construction and describe how the study of automorphism
groups has aided in the discovery of evidence supporting the conjectures. There are various
paradigms in which to view the results we summarize here. Different authors give their
results using their favorite formalism and rarely give more than a mention to the other
formalisms that have been used. Equiangular lines, SIC-POVMs and quantum designs
are all terms which represent the same object in different paradigms; here we attempt to
unify the known results by making explicit the correspondence between these objects (see
Figure 1.1) and interpreting the results using a single framework. (See Figure 1.2 for a
visual summary of the correspondence of results stated in the two major frameworks.)
In Chapter 3 we describe our new approaches to the construction of maximum-sized
sets of equiangular lines. In §3.1 we relate sets of equiangular lines to sets of mutually
unbiased bases formed from a relative difference set. This construction leads to a simplified
CHAPTER 1. OVERVIEW
6
interpretation of several known maximum-sized sets of equiangular lines in previously solved
dimensions. In §3.2 we display a previously unknown connection between equiangular lines
and complex Hadamard matrices. Using this connection, we give a new exact solution
in C8 which is simpler than any previously known solution in this dimension. In §3.3
we demonstrate a possible connection between maximum-sized sets of equiangular lines in
different dimensions by constructing another new exact solution for d = 8 from a d = 4
solution. In Chapter 4 we summarize our results and pose some questions that arise from
the results presented in Chapters 2 and 3.
7
CHAPTER 1. OVERVIEW
Figure 1.1: Thesis Cheat Sheet
(d, d, d, 1)Relative
Difference Set
d elements {rj } of
abelian group G of order d2 containing subgroup N of order d
such that
{rj rk−1 : j 6= k} = G\N
Theorem 17
Mutually
Unbiased
Bases
Complex
Hadamard
Matrix
d orthonormal bases
for Cd such that for
x, y in distinct bases
1
|hx, yi| = √
d
d × d matrix X with
complex entries of
magnitude 1 such
that
Theorem 38
X † X = dId
?
Theorem 49, 52, 54, 56
SIC-POVM
Equiangular
Lines
Quantum
Design
d2 projections d1 xj x†j
represented by unit
vectors xj such that
1
|hxj , xk i|2 =
d+1
for all j 6= k
d2 1-dimensional subspaces of Cd spanned
by unit vectors xj
such that
1
|hxj , xk i|2 =
d+1
for all j 6= k
d2 orthogonal projections Pj = xj x†j such
that tr(Pj ) = 1 for all
j, and
1
tr(Pj Pk ) =
d+1
for all j 6= k
Proposition 9
Proposition 9
Proposition 10
|
Numerical Solution
to machine precision
{z
Exact Solution
analytical expressions
}
Conjecture 25
Weyl-Heisenberg Covariant Conjecture 26
Conjecture 32
8
CHAPTER 1. OVERVIEW
Figure 1.2: Fiducial Construction Cheat Sheet
Vectors: L = {xj }
Designs: D = {Pj }
Group Covariant wrt G (2.40)
Group Covariant wrt G (2.41)
Group action is left multiplication
(i) L closed under action of G
(ii) G acts transitively on L
(up to phases)
Group action is conjugation
(i) D closed under action of G
(ii) G acts transitively on D
Pj = xj x†j
Proposition 23
Proposition 22
D = {AP1 A−1 : A ∈ G}
L = {αA Ax1 : A ∈ G}
Weyl-Heisenberg (2.38),(2.39)
φ
ω ` V j U k ∈ Hd
V j U k ∈ φ(Hd )
Conjecture 26, 32
There exist d2 equiangular lines in Cd for
d ≥ 2 of the form
(j, k) ∈ Zd × Zd
Conjecture 25
There exist maximal regular degree 1, r = 1
quantum designs for d ≥ 2 of the form
Pj = xj x†j
L = {αA Ax1 : A ∈ Hd }
Conjecture 30: can
take x1 = z
W
D = {AP1 A−1 : A ∈ Hd }
Fiducial Vector
Eigenvector z of Zu with
eigenvalue 1
Conjecture 28: can
take P1 = zz †
Zauner’s Unitary (2.44)
particular d × d unitary Zu
(preserves Weyl-Heisenberg
covariance)
Zu is an order 3 automorphism for L
of Conjecture 30
Zu ∈ Cd
Clifford Group (2.56)
Zu is an order 3 automorphism for D
of Conjecture 28
Cd is the normalizer of Hd
(preserves Weyl-Heisenberg
covariance, generates orbits
of fiducials)
Automorphism of L (2.52)
Unitary matrix A such that for
each j there exists αj ∈ S so
that
xj = αj Axπ(j)
for some permutation π
fiducial x
fiducial Ax
Pj = xj x†j
Automorphism of D (2.36)
Unitary matrix A such
that
Pj = APπ(j) A−1
for some permutation π
Chapter 2
Background and Previous Work
2.1
Equiangular Lines
Given two column vectors x, y ∈ Cd , define the inner product of x = (x1 , . . . , xd )T and
y = (y1 , . . . , yd )T by
hx, yi =
d
X
xj yj ,
(2.1)
j=1
where yj denotes the complex conjugate. Equivalently, letting x† be the conjugate transpose
xT of x, we have
hx, yi = x† y
(2.2)
xy † .
(2.3)
Define the outer product of x and y by
The norm of a vector x ∈ Cd is given by
||x|| =
Two vectors x, y are called orthogonal if
p
hx, xi.
hx, yi = 0.
(2.4)
A line through the origin in Cd can be represented by a vector x ∈ Cd which spans it.
Then the angle between two lines in Cd that are represented by vectors x, y is given by
|hx, yi|
arccos
.
||x|| · ||y||
9
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
10
We call a set of m ≥ 2 lines in Cd , represented by vectors x1 , . . . , xm , equiangular if there
is some constant c such that
arccos
|hxj , xk i|
||xj || · ||xk ||
= c,
for all j 6= k.
(2.5)
To simplify notation, we can always take x1 , . . . , xm to be unit vectors (vectors having norm
1), and then it suffices that there is a constant a < 1 such that
|hxj , xk i| = a,
for all j 6= k.
(2.6)
Example 4. Consider the lines in C4 represented by x1 = (1, 1, −1, −1)T , x2 = (1, i, −1, i)T ,
x3 = (1, −1, −i, −i)T . Then
|hx1 , x2 i|
|2|
arccos
= arccos
,
||x1 || · ||x2 ||
2·2
|hx1 , x3 i|
| − 2i|
arccos
= arccos
,
||x1 || · ||x3 ||
2·2
| − 2i|
|hx2 , x3 i|
= arccos
.
arccos
||x2 || · ||x3 ||
2·2
So the lines are equiangular with c = arccos(1/2). Since each vector has norm 2, we could
equivalently check that
D x x E
1 2 ,
=
2 2
D x x E
1 3 ,
=
2 2
D x x E
2 3 ,
=
2 2
1
,
2
1
,
2
1
.
2
Thus the unit vectors x1 /2, x2 /2, x3 /2 represent equiangular lines with a = 1/2.
Let X be a matrix defining a linear transformation from Cd to itself such that X 2 = X.
Then X is called a projection matrix. As in the case of vectors, write X † for the conjugate
transpose of matrix X. Then a complex matrix X for which
X† = X
(2.7)
is called a Hermitian matrix. If a projection matrix X is a Hermitian matrix, then it is
called an orthogonal projection. In particular, given a 1-dimensional subspace (line) in Cd ,
11
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
the orthogonal projection onto this line is given by a matrix X which is the outer product
of a unit vector x on the line with itself,
X = xx† .
(2.8)
In this thesis we will denote the d × d identity matrix by Id .
Example 5. Let ` be the line in C3 through the origin and (1, 0, i). Then ` can be represented
by the unit vector x =
√1 (1, 0, i)T
2

√1
2




X=
0


or by the orthogonal projection
√1
2
0 − √i2
√i
2
=

1 0 −i
1
 0 0
2
i 0


0 
.
1
(In the study of equiangular lines as SIC-POVMs, “bra-ket” notation is commonly used.
In this notation the vector x is written as |xi and x† is written as hx|. Using this notation
we can write the inner product hx, yi as hy|xi and the projection matrix xx† as |xihx|.)
The next propositions give useful relations between the outer product and the inner
product of vectors in Cd . Using these relations, in Theorem 8 we will be able to derive the
absolute upper bound for sets of equiangular lines in Cd .
Proposition 6. For vectors x, y ∈ Cd , tr(xy † ) = hx, yi.
Proof. The d × d matrix xy † satisfies
†
tr(xy ) =
d
X
(xy † )jj
j=1
=
d
X
xj yj
j=1
= hx, yi.
Proposition 7. Let X = xx† , Y = yy † be orthogonal projections. Then tr(XY ) = |hx, yi|2 .
Proof. By definition of X and Y we have
tr(XY ) = tr(xx† yy † )
12
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
= tr xhx, yiy †
= hx, yitr xy †
by (2.2)
since hx, yi is a scalar
= hx, yihx, yi
by Proposition 6
= |hx, yi|2 .
We can now present the absolute upper bound for sets of equiangular lines in Cd . This
upper bound gives the maximum number of equiangular lines possible based only on dimension. One can also give relative upper bounds, which depend on the dimension and a
predetermined angle between lines and which will be no greater than the absolute bound;
however, as we are concerned with the maximum number of equiangular lines in each dimension, we present only the absolute bound here.
Theorem 8 ([22]). A set of equiangular lines in Cd has size at most d2 .
Proof. Let x1 , . . . , xm be a set of unit vectors representing m equiangular lines in Cd . Let
Xj = xj x†j be the projection matrix associated to the line represented by xj for each j.
Then each Xj is a Hermitian matrix, as noted prior to (2.8).
Since every complex number can be represented by two real numbers, every d×d complex
matrix can be viewed as an element of a 2d2 -dimensional vector space V over R with basis
{Ejk , iEjk : j, k = 1, . . . , d},
where Ejk is the d × d matrix which has a single 1 in position (j, k) and zeros elsewhere.
The additional symmetry imposed by the definition of a Hermitian matrix means we can
write a d × d Hermitian matrix as a linear combination of the following matrices:
{Ejj : j = 1, . . . , d} ∪ {Ejk + Ekj , i(Ejk − Ekj ) : 1 ≤ j < k ≤ d}.
This is a set of d2 linear combinations of basis elements for V , which are clearly linearly
independent. Since these d2 elements span the set of Hermitian matrices, the subspace of
Hermitian matrices has dimension d2 ; thus there can be at most d2 linearly independent
Hermitian matrices. To prove the theorem all that remains is to show the {Xj } representing
the lines are linearly independent.
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
13
Since the xj are unit vectors,
hxj , xj i = 1.
(2.9)
Then we have
tr(Xj Xk ) = |hxj , xk i|2

a2 if j =
6 k
=
1
if j = k
by Proposition 7
(2.10)
for some constant a < 1 by (2.6), since x1 , . . . , xm are equiangular, and by (2.9). Next let
Yj = Xj − a2 Id ,
so that we have
tr(Yj Xk ) = tr((Xj − a2 Id )Xk )
= tr(Xj Xk ) − a2 tr(Xk )
= tr(Xj Xk ) − a2

0
if j =
6 k
=
1 − a2 if j = k,
by Proposition 6 and (2.9)
(2.11)
by (2.10), and 1 − a2 > 0 as a < 1. Finally, suppose we have some constants ck such that
0=
m
X
ck Xk .
k=1
Then we have
0 = tr Yj
m
X
ck Xk
k=1
=
m
X
!
ck tr(Yj Xk )
k=1
= cj (1 − a2 ),
by (2.11), which means that cj = 0 for each j. Thus the Xj are linearly independent, as
required.
Given a set of d2 equiangular lines in Cd it is possible to determine that the necessary
value of a2 , where a is given in (2.6), is
1
d+1 .
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
14
Proposition 9 ([22]). A set {xj } of d2 unit vectors in Cd represents a set of equiangular
lines if and only if
|hxj , xk i|2 =
1
,
d+1
for all j 6= k.
Proof. Notice we have just shown in the proof of Theorem 8 that the projection matrices
Xj = xj x†j representing a set of d2 equiangular lines {xj } form a basis for the space of
Hermitian matrices. This means we can write the d × d identity matrix as
2
Id =
d
X
ck Xk
k=1
for some ck ∈ C, and so
2
d = tr(Id ) =
d
X
ck ,
(2.12)
k=1
since tr(Xk ) = 1. Then for the Yj as defined in the proof of Theorem 8, we have
tr(Yj ) = tr(Yj Id )


d2
X
= tr Yj
ck Xk 
k=1
2
=
d
X
ck tr(Yj Xk )
k=1
which, by definition of the Yj and (2.11), gives
1 − a2 d = cj (1 − a2 ).
(2.13)
So from (2.13) we get that all the cj are equal, and from (2.12) their sum is d. Since there
are d2 of these coefficients, this means each cj is
solving for
a2
gives
a2 =
1
d.
Substituting this value in (2.13) and
1
.
d+1
Notice that using Proposition 7 we could write Proposition 9 as
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
15
Proposition 10. A set {Xj } of d2 orthogonal projections in Cd with Xj = xj x†j and
tr(Xj ) = 1 for all j, represents a set of equiangular lines if and only if
tr(Xj Xk ) =
2.2
1
,
d+1
for all j 6= k.
Mutually Unbiased Bases and Difference Sets
We will demonstrate connections between complex equiangular lines and other combinatorial
objects, including mutually unbiased bases and difference sets. We include the definitions
and some basic results about these objects here.
Let G be a group of order v with identity 1G . Then the elements of the group algebra of
G over C, written C[G], are the formal sums
X
zg g, where zg ∈ C.
g∈G
A (v, k, λ)-difference set in G is a subset D ⊂ G of size k, such that the multiset
{d1 d−1
2 : d1 , d2 ∈ D, d1 6= d2 }
contains each element of G\{1G } exactly λ times. Identifying a subset D ⊂ G with an
element of the group algebra C[G] as follows
D=
X
d,
d∈D
and writing D(−1) =
P
d∈D
d−1 we can say D is a (v, k, λ)-difference set if
DD(−1) = (k − λ)1G + λG
(in C[G]).
(2.14)
Example 11. Let G be the group hxi where x7 = 1. A (7, 3, 1)-difference set in G is given
by D = {1, x4 , x5 }, since
−4
−5 4
4 −5 5
5 −4
{d1 d−1
2 : d1 , d2 ∈ D, d1 6= d2 } = {1x , 1x , x 1, x x , x 1, x x }
= {x3 , x2 , x4 , x6 , x5 , x1 },
which is every element of G\{1} exactly λ = 1 times. The corresponding element of the
group algebra is D = 1 + x4 + x5 and
DD(−1) = (1 + x4 + x5 )(1 + x−4 + x−5 )
16
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
= 1 + x4 + x5 + x−4 + x0 + x1 + x−5 + x−1 + x0
= 2 + 1 + x4 + x5 + x3 + x1 + x2 + x6
= (3 − 1)1 + G.
Now let G be a group of order mn, containing a normal subgroup N of order n. A
(m, n, k, λ)-relative difference set (RDS) in G relative to N is a subset R ⊂ G of size k, such
that the multiset
{r1 r2−1 : r1 , r2 ∈ R, r1 6= r2 }
contains each element of G\N exactly λ times and does not contain any elements of N . As
with difference sets we can identify R ⊂ G with an element of the group algebra and say
that R is a relative difference set if
RR(−1) = k1G + λ(G − N )
(in C[G]).
(2.15)
Example 12. Let G be the abelian group of order 16 given by hxi × hyi, with x4 = y 4 = 1.
Let N be the subgroup hx2 i × hy 2 i of order 4. Then a (4, 4, 4, 1)-relative difference set is
given by R = {1, x, y, x3 y 3 }, since
{r1 r2−1 : r1 , r2 ∈ R, r1 6= r2 } = {1 · x−1 , 1 · y −1 , 1 · x−3 y −3 , x · 1, x · y −1 , x · x−3 y −3 , y · 1,
y · x−1 , y · x−3 y −3 , x3 y 3 · 1, x3 y 3 · x−1 , x3 y 3 · y −1 }
= {x3 , y 3 , xy, x, xy 3 , x2 y, y, x3 y, xy 2 , x3 y 3 , x2 y 3 , x3 y 2 }
which is every element of G\N exactly λ = 1 times. The corresponding element of the group
algebra is R = 1 + x + y + x3 y 3 and
RR(−1) = (1 + x + y + x3 y 3 )(1 + x−1 + y −1 + x−3 y −3 )
= 1 + x + y + x3 y 3 + x−1 + 1 + x−1 y + x2 y 3 + y −1 + xy −1 + 1 + x3 y 2
+ x−3 y −3 + x−2 y −3 + x−3 y −2 + 1
= 4 + x + y + x3 y 3 + x3 + x3 y + x2 y 3 + y 3 + xy 3 + x3 y 2
+ xy + x2 y + xy 2
= 4 · 1 + (G − N ).
Since the elements of the subgroup N do not appear as quotients of distinct elements of R,
N is referred to as the forbidden subgroup.
17
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
For any complex-valued function ψ on G, and D = {d1 , . . . , dk } ⊂ G, write
ψ(D) =
k
X
ψ(dj ).
(2.16)
j=1
Fix the ordering d1 , . . . , dk for the elements of the set D. Then represent the restriction of
ψ to D as a vector
ψ|D = (ψ(d1 ), . . . , ψ(dk ))T .
A complex-valued function χ : G → C is called a character of G if it is a group homo-
morphism. When G is a finite abelian group of order v, there are exactly v characters of G,
χ1 , . . . , χv , where χ1 denotes the principal character,
χ1 (g) = 1,
for all g ∈ G,
and the remaining characters are called non-principal. It is not hard to show that every χj
must map G to the m-th roots of unity, where m is the exponent of G. Furthermore, if we
have a subgroup H ⊆ G, then each character χj for which
χj (h) = 1,
for all h ∈ H,
is called principal on H. We can show that for all characters χj which are non-principal on
H,
χj (H) = 0.
(2.17)
The characters of G form a group under multiplication, where for two characters χj , χk
(χj χk )(g) = χj (g)χk (g),
for all g ∈ G.
(2.18)
Denote this group of characters by G∗ . It can be shown that G∗ is isomorphic to the group
G. Using this isomorphism to label the characters of G by the elements of G, we have
χa χb = χab
for all a, b ∈ G,
(2.19)
and for each character χa , we have the following relations for all g ∈ G
1 =
χ1G (g)
=
χaa−1 (g)
1 = |χa (g)|2
1 =
χa (1G )
= χa (g)χa−1 (g)
by (2.19),
= χa (g)χa (g)
as χa (g) is a root of unity,
= χa (gg −1 ) = χa (g)χa (g −1 )
as χa is a homomorphism.
So we see that for all g ∈ G,
−1
χ−1
a (g) = χa−1 (g) = χa (g) = χa (g ).
(2.20)
18
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Example 13. Let G = hxi ∼
= Z4 . Now G has exponent 4, so the characters map G to the
fourth roots of unity. Furthermore G has 4 characters, G∗ = {χj : j ∈ Z4 }, where
χj (x) = ij .
Since each χj is a homomorphism and x generates G, the value χj (x) determines χj completely via χj (xk ) = (χj (x))k . Notice that since we label characters by elements of Z4 , 0 is
the identity, so that χ0 = χ1G is the principal character in this example.
Let X, Y be d×d matrices with columns {xj }dj=1 , {y j }dj=1 , respectively. Then the matrix
of (conjugated) inner products between each column of X and each column of Y is given by
X † Y = (x†j y k )dj,k=1 .
(2.21)
A matrix X is called a complex Hadamard matrix if it has all entries in C and of magnitude
1 and its columns are pairwise orthogonal. Equivalently, using (2.21), X is a d × d complex
Hadamard matrix if
X † X = dId .
(2.22)
If all the entries of X are in {1, −1} then X is called a real Hadamard matrix or just a
Hadamard matrix.
A basis for Cd is called orthogonal if every two distinct basis elements are orthogonal.
Let B be a d × d matrix with all entries in C and of magnitude 1. By (2.21), the columns
√
of B are the elements of an orthogonal basis each having norm d if and only if B is a
complex Hadamard matrix.
Example 14. Consider the orthogonal basis {(1, i)T , (1, −i)T } for C2 . Then
!
1 1
B=
i −i
and
†
B B =
=
so B is a complex Hadamard matrix.
1 −i
1
i
2 0
0 2
!
!
,
1
1
i −i
!
19
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Let {x1 , . . . , xd }, {y 1 , . . . , y d } be two distinct orthogonal bases for Cd . They are called
unbiased bases if for all xj , y k
|hxj , y k i|
1
=√ .
||xj || · ||y k ||
d
√
In particular, if all basis elements have norm d, then
|hxj , y k i| =
√
(2.23)
d.
(2.24)
A set of orthogonal bases is a set of mutually unbiased bases (MUBs) if all pairs of distinct
bases are unbiased.
Example 15. Consider the following orthogonal bases for C2 :
)
)
)
(
(
(
(1 0)T
(1 1 )T
(1 i )T
B2 =
B3 =
.
B1 =
(0 1)T
(1 −1)T
(1 −i)T
Notice all elements of B1 have norm 1 and all elements of B2 , B3 have norm
√
2. The inner
product of any element of B1 with any element of B2 or B3 has magnitude 1. The inner
√
product of an element of B2 with an element of B3 is 1 ± i, which has magnitude 2. Thus
for x, y in distinct bases we have
|hx, yi|
||x|| · ||y||
=
=


1
√
1· 2
√
√ √
2
2· 2
for one of x, y ∈ B1
for x, y ∈
/ B1
1
√ ,
2
satisfying (2.23), so {B1 , B2 , B3 } is a set of MUBs in C2 .
Proposition 16 ([8, Theorems 3.4, 3.5]). A set of MUBs in Cd has size at most d + 1.
Proof. Let B1 , . . . , Bm be a set of MUBs in Cd . Let Bj = {xj1 , . . . , xjd }, where we assume
by suitable normalization that each ||xjk || = 1. Let δq,r be the Kronecker delta function.
Then by (2.23) we have

δq,r
j
k
hxq , xr i =
q,r
 eiθ
√
d
for some real constants θq,r , depending on j, k, q, r.
if j = k
if j 6= k,
(2.25)
20
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Now define matrices
Uj,t =
d
X
e2πi(tk/d) xjk (xjk )† ,
k=1
for 1 ≤ j ≤ m and 0 ≤ t ≤ d − 1.
(2.26)
Suppose that
Uj,t = U`,s for some (j, t) 6= (`, s).
(2.27)
We will show that we must have s = t = 0, which means that we have defined at least
m(d − 1) + 1 distinct matrices Uj,t .
From (2.27), for all 1 ≤ r ≤ d we have
Uj,t x`r = U`,s x`r
(2.28)
so
d
X
k=1
e2πi(tk/d) xjk hxjk , x`r i =
=
=
d
X
e2πi(tk/d) xjk (xjk )† x`r
by (2.2)
e2πi(sk/d) x`k (x`k )† x`r
by (2.28) and (2.26)
e2πi(sk/d) x`k hx`k , x`r i
by (2.2)
k=1
d
X
k=1
d
X
k=1
= e2πi(sr/d) x`r
by (2.25).
(2.29)
First, suppose j = `, then by (2.25) we have that (2.29) becomes
e2πi(tr/d) x`r = e2πi(sr/d) x`r .
In other words, we must have e2πi(tr/d) = e2πi(sr/d) for all r, which is only possible if s = t.
This means (j, t) = (`, s), which contradicts our assumption (2.27).
Now suppose j 6= `. Then by (2.25) we have that (2.29) becomes
d
X
eiθk,r
e2πi(tk/d) xjk √ = e2πi(sr/d) x`r ,
d
k=1
and for each 1 ≤ q ≤ d, multiplying on the left by (xjq )† gives
d
X
eiθk,r
e2πi(tk/d) (xjq )† xjk √ = e2πi(sr/d) (xjq )† x`r ,
d
k=1
21
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
which using (2.2) we can write as
d
X
eiθk,r
e2πi(tk/d) hxjq , xjk i √ = e2πi(sr/d) hxjq , x`r i.
d
k=1
Using (2.25), this becomes
eiθq,r
eiθq,r
e2πi(tq/d) √ = e2πi(sr/d) √ ,
d
d
which implies
e2πi(tq/d) = e2πi(sr/d)
for all q, r, which forces s = t = 0.
So we have defined at least m(d − 1) + 1 distinct d × d complex matrices. We now show
that all these matrices are pairwise orthogonal, where two matrices A = (ajk ), B = (bjk )
P
P
are orthogonal if tr(A† B) = 0. Notice that tr(A† B) = dj=1 dk=1 bjk ajk , which is simply
2
the inner product of two vectors b, a ∈ Cd whose components are the entries of B and A,
2
respectively. There can be at most d2 orthogonal vectors in Cd ; thus we can have at most
d2 orthogonal d × d matrices. Thus m(d − 1) + 1 ≤ d2 , which implies m ≤ d + 1, proving
the proposition.
It remains to show that
†
tr(Uj,t
U`,s ) = 0,
First let Vq,j = xjq (xjq )† . Then by (2.26)

†
d
X

†
tr(Uj,t
U`,s ) = tr 
e2πi(tq/d) Vq,j 
q=1
for (s, t) 6= (0, 0).
d
X
r=1
!
(2.30)


e2πi(sr/d) Vr,` 


d X
d
X
= tr 
e2πi(sr−tq)/d Vq,j Vr,` 
since Vq,j is Hermitian
q=1 r=1
=
d X
d
X
e2πi(sr−tq)/d tr(Vq,j Vr,` )
q=1 r=1
=
d X
d
X
q=1 r=1
e2πi(sr−tq)/d |hxjq , x`r i|2
by Proposition 7.
22
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Now when j = ` and (s, t) 6= (0, 0) we have
†
tr(Uj,t
Uj,s ) =
d X
d
X
q=1 r=1
=
d X
d
X
e2πi(sr−tq)/d |hxjq , xjr i|2
e2πi(sr−tq)/d δq,r ,
by (2.25)
q=1 r=1
=
d
X
e2πi(s−t)q/d
q=1
= e2πi(s−t)/d
= 0,
1 − (e2πi(s−t)/d )d
1 − e2πi(s−t)/d
by summing the geometric series
since s − t 6= 0, so that 1 − e2πi(s−t)/d 6= 0.
Finally if j 6= ` and (s, t) 6= (0, 0), then
†
tr(Uj,t
U`,s ) =
d X
d
X
q=1 r=1
=
d X
d
X
e2πi(sr−tq)/d |hxjq , x`r i|2
e2πi(sr−tq)/d
q=1 r=1

1
= 
d
= 0,
d
X
q=1

e2πi(tq/d) 
1
d
by (2.25)
d
X
r=1
e2πi(sr/d)
!
since at least one of s, t is nonzero, so that one of the sums is again a geometric series
summing to 0.
A set of d + 1 MUBs in Cd is called complete. Following the method of Godsil and Roy
[23], we now show a construction for a complete set of MUBs from an RDS.
Theorem 17 (Theorem 4.1, [23]). The existence of a (d, d, d, 1)-RDS in an abelian group
implies the existence of a set of d + 1 MUBs in Cd .
Proof. Let R = {r1 , . . . , rd } be a (d, d, d, 1)-RDS in an abelian group G relative to a subgroup
N . Let G∗ be the group of characters of G and N ⊥ be the subgroup of characters which
are principal on N which are induced by the characters of G/N . Now |G∗ | = |G| = d2 and
23
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
|N ⊥ | = |G/N | = d, so there are |G∗ /N ⊥ | = d cosets of N ⊥ in G∗ . Let χ1 , . . . , χd be the
coset representatives of N ⊥ in G∗ , with χ1 being principal on G. Put the restriction of each
of the characters in G∗ to R into d blocks of d vectors as follows:
Bj = {(χj ψ)|R : ψ ∈ N ⊥ } = {(χj ψ(r1 ), χj ψ(r2 ), . . . , χj ψ(rd ))T : ψ ∈ N ⊥ }.
Consider the magnitude of the inner product of two of the above vectors from blocks Bj
and Bk , namely
d
X
|h(χj ψ)|R , (χk φ)|R i| = χj ψ(rn )χk φ(rn )
n=1
d
X
=
χj ψ(rn )φ−1 χ−1
(r
)
n
k
n=1
d
X
=
(r
)
χj ψφ−1 χ−1
n k
n=1
d
X
=
χ(rn )
by (2.20)
by (2.18)
where χ = χj ψφ−1 χ−1
k
n=1
= |χ(R)|
by (2.16).
(2.31)
Now R is an RDS and so
|χ(R)|2 = χ(R)χ(R)
= χ(R)χ(R(−1) )
= χ(RR(−1) )
= χ(d1G + (G − N ))

0 if χ ∈ N ⊥ \{χ1 }
=
d if χ ∈
/ N ⊥,
by (2.20)
since χ is a homomorphism
by (2.15) with λ = 1
(2.32)
by (2.17) and since χ ∈ N ⊥ \{χ1 } is principal on N and |N | = d.
Thus if we take two distinct vectors from the same block Bj , then ψ 6= φ and since G∗
−1 which is in N ⊥ \{χ }, since ψ, φ are distinct characters
is abelian χ = χj ψφ−1 χ−1
1
j = ψφ
in N ⊥ . Then by (2.31) and (2.32) the magnitude of their inner product is 0. Otherwise we
have two vectors from different blocks and χ = χj ψφ−1 χ−1
/ N ⊥ so that the magnitude of
k ∈
√
their inner product is d by (2.31) and (2.32). Since every character maps G to the m-th
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
roots of unity for some m, each (χj ψ)|R has norm
√
24
d, so that by (2.24) our d blocks of d
vectors actually form d MUBs.
Notice that using (2.23) it is easy to check that each Bj is also unbiased with respect to
the standard basis for Cd for a total of d + 1 MUBs, as required.
Example 18. Recall the RDS from Example 12, R = {1, x, y, x3 y 3 } in G = hxi × hyi ∼
=
Z4 × Z4 . The characters of G map the elements of G to the fourth roots of unity, so that
G∗ = {χj,k : (j, k) ∈ Z4 × Z4 }, where
χj,k (x) = ij ,
χj,k (y) = ik .
Then χ0,0 is principal on G and the characters which are principal on the forbidden subgroup
N = hx2 i×hy 2 i ∼
= Z2 ×Z2 are given by N ⊥ = {χj,k : (j, k) ∈ {(0, 0), (0, 2), (2, 0), (2, 2)}} and
the 4 coset representatives χ1 , χ2 , χ3 , χ4 are given by χ0,0 , χ0,1 , χ1,0 , χ1,1 . Then the MUBs
constructed as in Theorem 17, are formed as follows:
(1
x
y
χ0,0 (1
1
1
χ0,2 (1
1
−1
χ2,0 (1 −1
1
χ2,2 (1 −1 −1
χ0,1 (1
1
i
χ0,3 (1
1
−i
χ2,1 (1 −1
i
χ2,3 (1 −1 −i
χ1,0 (1
i
1
χ1,2 (1
i
−1
χ3,0 (1 −i
1
χ3,2 (1 −i −1
χ1,1 (1
i
i
χ1,3 (1
i
−i
χ3,1 (1 −i
χ3,3 (1 −i
i
−i
x3 y 3 )T

1 )T 



−1 )T 
B1
−1 )T 




1 )T

−i )T 



i )T 
B2
i )T 




−i )T

−i )T 



T
i ) 
B3

i )T 



−i )T

−1 )T 



T
1 ) 
B4
1 )T 




−1 )T
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
25
Each χj,k |R has norm 2 and one can check that any two vectors x, y from distinct bases
√
Bj , Bk satisfy hx, yi ∈ {2, 2i, −2i}, which has magnitude 2 = 4, as required by (2.24).
Furthermore it is easy to see that the inner product of each element of the standard basis
with each element of every Bj has magnitude 1, so that (2.23) is also satisfied.
It is known that one can construct an RDS of the form necessary for Theorem 17 in
r
∼ r
groups G = Z2r
p relative to N = Zp for p an odd prime, r ≥ 1, and in G = Z4 relative to
N∼
= Zr , r ≥ 1 [34]; thus the bound of Proposition 16 can be attained with equality when d
2
is a prime power. When d is not a prime power, the smallest dimension for which a set of
d + 1 MUBs is not known is d = 6. Many sets of 3 MUBs in C6 are known, but no one has
found even a set of 4 MUBs. Some researchers even suspect that it may not be possible to
find more than 3 MUBs in C6 [10].
The following result, similar to Theorem 17 and also given in [23], explores a connection
between equiangular lines and difference sets.
Theorem 19 (Theorem 2.1, [23]). The existence of a (v, k, 1)-difference set in an abelian
group implies the existence of v = k 2 − k + 1 equiangular lines in Ck .
Proof. Let D = {d1 , . . . , dk } be a (v, k, 1)-difference set in an abelian group G of order v.
Since the elements of a (v, k, 1)-difference set form 2 k2 differences which cover each of the
v − 1 elements of G\{1G } exactly once, we get
k
2
=v−1
2
and so
v = k 2 − k + 1.
Let G∗ be the group of characters χ1 , . . . , χv , where χ1 is principal on G. Form the vectors
which are the restrictions of the characters to D, namely
{χ|D : χ ∈ G∗ } = {(χ(d1 ), . . . , χ(dk ))T : χ ∈ G∗ }.
Consider the magnitude of the inner product of two of the above vectors, namely
d
X
χj (dn )χ` (dn )
|hχj |D , χ` |D i| = n=1
d
X
=
χj (dn )χ`−1 (dn )
by (2.20)
n=1
26
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
d
X
=
χj χ`−1 (dn )
n=1
d
X
=
χ(dn )
by (2.18)
where χ = χj`−1 by (2.19)
n=1
= |χ(D)|
by (2.16).
(2.33)
Now D is a difference set and so for χ non-principal on G,
|χ(D)|2 = χ(D)χ(D)
= χ(D)χ(D(−1) )
= χ(DD(−1) )
= χ((k − 1)1G + G)
by (2.20)
since χ is a homomorphism
by (2.14) with λ = 1
= k − 1,
(2.34)
by (2.17) since χ is non-principal. Now χ = χj`−1 is non-principal when j`−1 6= 1G ; thus by
(2.33) and (2.34) we have
|hχj |D , χ` |D i|2 = k − 1,
for all j 6= `
so that the v = k 2 − k + 1 restrictions of the characters to D are equiangular.
Difference sets with λ = 1 are known as planar difference sets. It is known that one can
construct difference sets of the form required for Theorem 19 for G = Zv and k = 1 + q
for q a prime power (a Singer difference set) [13]. The Prime Power Conjecture states that
if an abelian planar difference set exists, then q is a prime power. The construction of
Theorem 19 gives v = k 2 − k + 1 equiangular lines in Ck , which is not a set of maximum
size. Furthermore it can be shown that k 2 − k + 1 is the maximum number of flat (all vector
entries having the same magnitude) equiangular lines possible in Ck [23, Lemma 2.2].
We now consider the work that has already been done in finding maximum-sized sets
of equiangular lines. We detail the major advances this area, identifying what we consider
to be key examples and landmark papers. Since equiangular lines are studied under several
guises we also attempt to unify the known results by interpreting them all from a single
viewpoint.
We begin with two key examples of maximum-sized sets of equiangular lines.
27
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
2.3
A Simple Example of 16 Equiangular Lines for d = 4
In his thesis [9, p. 51], Belovs gives an example of 16 equiangular lines in C4 which have a
particularly simple form. We

m n
n
n
m

 n im
n
in −n


 n −in −im n
n

n n
in −im −n
where
list them here as the columns of the following matrix:
n
n
n
m
n
n
−im
−n
−in
n
im
n
1
m=
2
s
n
m
n
n
n


in −n −im −n −in 

,
−in −im n −n in im −n −n in im −n 

−n −in im −n −n −in im n
n
in −im
1
n=
2
s
1
1− √ ,
5
3
1+ √ .
5
It is not hard to see that the inner product of any 2 distinct columns has magnitude |m2 −n2 |
or |mnz ± n2 z|, where z ∈ {1 + i, 1 − i}. Then the magnitude of the inner product of any
two distinct vectors is either
2
2
|m − n | =
=
1
3
1
1
1+ √
1− √
−
4
4
5
5
1
√
5
or |mn ± n2 + t(mn ∓ n2 )i| for t ∈ {1, −1}, namely,
p
(mn ± n2 )2 + (mn ∓ n2 )2 =
=
=
=
=
=
√ p
2 n2 (m2 + n2 )
s √
1
1
1
1
3
1
+
2
1− √
1+ √
1− √
4
4
4
5
5
5
√ s
2
1
2
1− √
2+ √
4
5
5
√ r
2
2
2−
4
5
r
1 16
4 5
1
√ .
5
Thus these vectors satisfy the conditions of Proposition 9.
28
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
2.4
Hoggar’s 64 Equiangular Lines for d = 8
Hoggar [30] constructs two quaterionic polytopes. He notes in his concluding remarks that
the 64 diameters of his first polytope constitute a set of lines in quaternionic 4-space having
angles with squared cosines
1
3
and
1
9.
He also states that upon complexification of these
lines, one obtains a set of 64 equiangular lines in C8 attaining the bound of Theorem 8, a
fact which he explains further in [32].
In his thesis [42, p. 65], Zauner gives a direct construction of Hoggar’s lines in complex
space without considering Hoggar’s quaternionic polytope. This set of 64 equiangular lines in
C8 is considered again later by Godsil and Roy [23]. They give the following construction for
Hoggar’s lines, which we will see at the end of §2.5.1 is a variation of Zauner’s construction.
Consider the following matrices:
X=
0 1
1 0
!
,
Z=
1
0
0 −1
!
.
These are known as the Pauli matrices (or generalized spin matrices) and they generate a
group of order 4, hX, Zi/h−I2 i sometimes known as the Pauli group. Let G be the group
(modulo h−I2 i) which is the 3-fold tensor product of the matrices in the Pauli group. Letting
Y = XZ, we can write this group as G = {I2 , X, Y, Z}⊗3 . Then G has order 64, and letting
1 + i 1 − i 1 + i −(1 + i) √
, 0, 2 ,
x = 0, 0, √ , √ , √ , √
2
2
2
2
we get that {M x : M ∈ G} is a set of 64 vectors in C8 . Careful consideration of the inner
products of vectors in this set reveals that this is, in fact, a set of equiangular lines.
It can be shown that the obvious generalization of this construction, taking G = {I, X, Y, Z}⊗k
for any k, gives equiangular lines only for k = 1, 3 [23]. However, this example illustrates how
even a small variation on the standard construction can yield a considerable simplification
to a maximum-sized set of equiangular lines (see §A.1).
2.5
Zauner’s Conjecture
Recall Conjecture 2 made by Zauner in his 1999 thesis [42, p. 59]:
Conjecture 2. For each integer d ≥ 2, there is a set of d2 equiangular lines in Cd .
29
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Zauner made this conjecture having only his own exact solutions for d = 2, 3, 4, 5, knowledge of Hoggar’s exact d = 8 solution and numerical solutions for d = 6, 7. This was the first
time such a conjecture was made about complex equiangular lines. The conjecture remains
open today, though the amount of evidence in its favour has grown significantly since it
was first proposed by Zauner. Zauner’s thesis is foundational to the study of equiangular
lines today as the construction he describes has since become the standard construction for
maximum-sized sets of equiangular lines. In what follows we look at his conjecture in more
detail and describe his construction method.
2.5.1
Group-Covariance (A First Refinement of Zauner’s Conjecture)
In his thesis [42], Zauner investigates quantum designs. In particular he explores in detail
the construction of maximal regular quantum designs with degree 1, r = 1. Zauner defines
a quantum design as a set D = {P1 , . . . , Pm } of d × d orthogonal projection matrices. The
design is regular if tr(Pj ) = r for all 1 ≤ j ≤ m. The cardinality of the set {tr(Pj Pk ) : 1 ≤
j 6= k ≤ m} is called the degree of the design and the design is called maximal if D contains
d2 elements.
These maximal designs, in fact, correspond to maximum-sized sets of equiangular lines
(see Figure 1.1). Since the trace of a projection matrix gives the dimension of the subspace
onto which it projects, if the design D = {P1 , . . . Pd2 } is regular with r = 1, then the elements
Pj are orthogonal projections onto lines in Cd (in particular, each Pj can be written as xj x†j
for some unit vector xj spanning the subspace). The fact that the design has degree 1
means that {tr(Pj Pk ) : Pj , Pk ∈ D, j 6= k} = {a2 } for some a. Then by Proposition 7 and
(2.6) the lines represented by the projections are equiangular.
Example 20. Let D = {P1 , P2 , P3 }, where ω = e2πi/3



1 ω2 0
1 0 ω2



1

P1 = 21 
 ω 1 0  P2 = 2  0 0 0
0 0 0
ω 0 1
and


 P3 =

1
2

0 0 0



 0 1 1 .


0 1 1
It is easy to check that Pj2 = Pj and Pj† = Pj for each j. Then D is a regular design with
r = 1, as tr(Pj ) = 1 for all j. Furthermore, one can check that tr(Pj Pk ) = 1/4 for j 6= k,
so the design has degree 1. It is not a maximal design, as it has only 3 elements rather than
32 = 9.
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
30
A d × d matrix X is called unitary if XX † = Id ; equivalently, X is unitary if
X −1 = X † .
(2.35)
An automorphism of a quantum design D = {P1 , . . . , Pm } is given by a unitary matrix A
such that
Pj = APπ(j) A−1
for all 1 ≤ j ≤ m,
(2.36)
for some permutation π ∈ Sm . The automorphisms of a design form a group. If there
is a subgroup G of the automorphism group such that the group of permutations induced
by elements of G acts transitively on {1, . . . , m}, then for each 1 ≤ j ≤ m there exists a
permutation πj , corresponding to some Aj ∈ G, such that πj (j) = 1 so that
Pj = Aj P1 A−1
j .
−1 : A ∈ G}.
So we can see that D = {Pj : 1 ≤ j ≤ m} = {Aj P1 A−1
j : 1 ≤ j ≤ m} ⊆ {AP1 A
However, G contains only automorphisms of D so that for each A ∈ G, by (2.36) AP1 A−1 =
Pπ−1 (1) ∈ D for some π ∈ Sm . Thus {AP1 A−1 : A ∈ G} ⊆ D and we may write
D = {AP1 A−1 : A ∈ G}.
(2.37)
A subgroup of the automorphism group is called regular if the group of permutations it
induces is transitive and each non-identity element has no fixed points. It is known that
such a subgroup must have order m [13], and thus is a minimal subgroup for which we can
write D in the form (2.37).
Example 21. Let

0

A=
 0
ω2
1 0


0 ω 
.
0 0
Then it is not hard to check that for D from Example 20
P1 = AP3 A−1 , P2 = AP1 A−1 , P3 = AP2 A−1 ,
so that A is an automorphism of D inducing permutation π = [3, 1, 2]. Furthermore, one can
check that the group G generated by A, namely G = {I3 , A, A2 }, consists of automorphisms
of D. In fact, G is a regular subgroup of the automorphism group, since it induces permutations {[1, 2, 3], [3, 1, 2], [2, 3, 1]}, which act transitively on {1, 2, 3} and only the identity
permutation has any fixed points. Thus we can write D = {AP1 A−1 : A ∈ G}.
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
31
We now consider a particular subgroup of the group of unitary matrices. Let ω = e2πi/d
and define U, V to be the following d × d matrices:

1 0 0 ···

 0 ω 0 ···


2
U =
 0 0 ω ···
 ..
..
 .
.

0 0 0 ···
and






V =





0 1 0
0
0
0
..
.
ω d−1
···
0 0 1 0 ···
..
..
.
.
0 0 0
0
1 0 0
0

0
···
···
0




,










.



1 

0
0
..
.
It is not hard to check that V U = ωU V . Then the group hU, V i has d3 elements given by
ω ` V j U k for `, j, k ∈ {0, 1, . . . , d−1}; it is known as the Weyl-Heisenberg (or sometimes simply
Heisenberg) group [41]. Denote this group by Hd . Its center is Z(Hd ) = {Id , ωId , . . . ω d−1 Id }.
Let φ be the canonical map from Hd to Hd /Z(Hd ), given by
φ : Hd → Hd /Z(Hd )
h 7→ hZ(Hd ).
(2.38)
The group Hd /Z(Hd ) has d3 /d = d2 elements which consist of all possible products of
U and V , modulo phases. Since V U = ωU V , this means we can write every element of
φ(Hd ) = Hd /Z(Hd ) as V j U k for some j, k ∈ {0, 1, . . . , d − 1}; thus Hd /Z(Hd ) ∼
= Zd × Zd .
The matrices V j U k are known as Weyl matrices. Let W be the isomorphism from Zd × Zd
to Hd /Z(Hd ) given by W (j, k) = V j U k , so that
φ(ω ` V j U k ) = W (j, k).
(2.39)
(See Figure 1.2).
Recall in Chapter 1 we introduced the notion of group covariance for a set of lines L,
which means that the set of lines is closed under the action of the group and the group acts
transitively on the set of lines. We now make that definition more precise. Let S = {z ∈ C :
32
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
|z| = 1}. A set of vectors L (representing lines in Cd ) is called group covariant if there is a
group G with some unitary representation using d × d matrices {Ug }g∈G , such that
(i)
for each Ug and each x ∈ L there is some α ∈ S such that Ug x ∈ αL,
(ii) for each pair xj , xk ∈ L there exists Ug such that xj = αUg xk for some α ∈ S.
(2.40)
If we know the particular group G for which this holds, then we can say that L is group
covariant with respect to G or that L is G covariant.
Notice that in this definition of group covariant for vectors, we allow the group to map
the vectors to the same set of vectors up to phases. This is because if unit vectors {xj }
represent a set of maximum-sized equiangular lines in Cd , then so do {αj xj } where each
αj ∈ S. To see this notice that for j 6= k,
|hαj xj , αk xk i|2 = |hxj , xk i|2
1
=
d+1
by (2.1) and since αj , αk ∈ S
by Proposition 9.
An analogous definition holds for quantum designs, where since designs are sets of matrices instead of vectors we will consider the action of the group to be conjugation instead
of left multiplication. We say that a quantum design D of d × d matrices is group covariant
if there is a group G with some unitary representation using d × d matrices {Ug }g∈G , such
that
(i)
for each Ug and each P ∈ D, we have Ug P Ug−1 ∈ D,
(ii) for each pair Pj , Pk ∈ D there exists Ug such that Pj = Ug Pk Ug−1 .
(2.41)
To see that the definition of group covariant for vectors implies the definition for the
associated designs, for each vector x ∈ L form the associated projection matrix X = xx†
and let D be the set {X : x ∈ L} (see Figure 1.2). Then (i) of (2.40) tells us for each Ug
and x ∈ L that Ug x = αy for some y ∈ L. This means that for each Ug and each X ∈ D,
Ug XUg−1 = Ug xx† Ug†
since Ug unitary
= Ug x(Ug x)†
= αy(αy)†
= yy †
since α ∈ S
and this is an element of D, since y ∈ L, so that (i) of (2.41) holds. Then (ii) of (2.40) tells
us that for each pair xj , xk ∈ L there exists Ug such that xj = αUg xk for some α ∈ S. This
33
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
means that for each pair Xj , Xk ∈ D there exists Ug such that
Xj = xj x†j = αUg xk (αUg xk )†
= αUg xk x†k Ug−1 α
since Ug is unitary and α is a scalar
= Ug Xk Ug−1
since α ∈ S
so that (ii) of (2.41) also holds.
An analogous argument shows that (2.41) implies (2.40) by noting that
xk x†k = Ug xj (Ug xj )† ⇔ αxk = Ug xj
for some α ∈ S.
We have seen that if we have a transitive subgroup of the automorphism group of a
design, then we can generate the design using this group and a single element of the design.
We now see that this is equivalent to group covariance.
Proposition 22. A design D is group covariant with respect to G if and only if we can
write D = {AP1 A−1 : A ∈ G} for some P1 ∈ D.
Proof. It is not hard to show that D of the form (2.37) is group covariant with respect to G.
Conversely, if D is group covariant with respect to G then (i) of (2.41) gives {AP1 A−1 : A ∈
G} ⊆ D. For each j, (ii) of (2.41) tells us that there is an Aj ∈ G such that Pj = Aj P1 A−1
j ,
−1 : A ∈ G}.
so that D = {Pj : 1 ≤ j ≤ m} = {Aj P1 A−1
j : 1 ≤ j ≤ m} ⊆ {AP1 A
We restate Proposition 22 in terms of vectors (see Figure 1.2):
Proposition 23. A set of vectors L is group covariant with respect to G if and only if we
can write
L = {αA Ax1 : A ∈ G}
(2.42)
for some x1 ∈ L and some constants αA ∈ S.
Proof. To see that L of the form (2.42) is group covariant with respect to G, notice that for
−1
each B ∈ G and αA Ax1 ∈ L, there exists β = αA αBA
∈ S such that
B(αA Ax1 ) = β(αBA BAx1 ) ∈ βL
and (i) of (2.40) holds. For each pair αA Ax1 , αB Bx1 ∈ L, there exists C = AB −1 ∈ G and
−1
β = αA αB
∈ S such that
αA Ax1 = βC(αB Bx1 )
34
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
so that (ii) of (2.40) is satisfied.
Conversely, if L is group covariant with respect to G then (i) of (2.40) gives {αA Ax1 :
A ∈ G} ⊆ L. For each j, (ii) of (2.41) tells us that there is an Aj ∈ G and αj ∈ S such that
xj = αj Aj x1 , so that L = {xj : 1 ≤ j ≤ m} = {αj Aj x1 : 1 ≤ j ≤ m} ⊆ {αA Ax1 : A ∈ G}.
Example 24. Let D = {P1 , P2 , P3 } be the design from



1 ω2 0
1 0 ω2



1

P1 = 21 
 ω 1 0  , P2 = 2  0 0 0
0 0 0
ω 0 1
and G = {I3 , A, A2 } be the
 


  1
 0
G=




0
Example 20



 , P3 =

1
2
0 0 0



 0 1 1 


0 1 1
regular subgroup from Example 21
 
 
 

0 0
0 1 0
0 0 ω

 
 
 

 

1 0 
,  0 0 ω ,  1 0 0  .


0 1
ω2 0 0
0 ω2 0
Then in Example 21 we have seen that
D = {AP1 A−1 : A ∈ G},
so that D is group covariant with respect to G, by Proposition 22.
Furthermore, this corresponds to a set of 3 equiangular lines in C3 . To see this, one can
check that Pj = xj x†j for
x1 =
x2 =
x3 =
1
√ (1, ω, 0)T
2
1
√ (1, 0, ω)T
2
1
√ (0, ω, ω)T ,
2
and clearly, the magnitude of the inner product of any two distinct vectors is 1/2. Now for
automorphism A ∈ G from Example 21, we have the following:
Ax1 = ωx2
Ax2 = ωx3
Ax3 = ωx1
35
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
A2 x 1 = ω 2 x 3
A2 x 2 = ω 2 x 1
A2 x 3 = ω 2 x 2 .
Thus the lines satisfy (2.40), making them group covariant with respect to G = {I3 , A, A2 }.
By Proposition 23 this means we can write
L = {xj : 1 ≤ j ≤ 3} = {αA Ax1 : A ∈ G},
where we can see that αId = 1, αA = ω −1 , αA2 = ω −2 .
In the case of the Weyl-Heisenberg group, if we write D = {AP A−1 : A ∈ Hd }, for some
initial projection P , then it appears that D has d3 (more than maximal) elements; however,
notice that since the elements of Hd are {ω ` V j U k : `, j, k ∈ Zd }, the elements of D are
{AP A−1 : A ∈ Hd } = {ω ` V j U k P (ω ` V j U k )−1 : `, j, k ∈ Zd }
= {ω ` V j U k P U −k V −j ω −` : `, j, k ∈ Zd }
= {V j U k P U −k V −j : j, k ∈ Zd }
= {AP A−1 : A ∈ φ(Hd )},
by (2.39).
Thus D actually has only d2 distinct elements, given by the action of the Weyl matrices on
the initial projection. Therefore, we can write
D = {AP A−1 : A ∈ φ(Hd )}
(2.43)
and still say that D is Weyl-Heisenberg covariant. Such a design has Hd as a transitive
subgroup of its automorphism group, and φ(Hd ) as a regular subgroup. Furthermore, since
W is an isomorphism between Zd × Zd and φ(Hd ), we can view φ(Hd ) as a representation of
Zd × Zd ; thus some authors also say that D satisfying (2.43) is group covariant with respect
to Zd × Zd .
In §2.5.2 we will see another conjecture made by Zauner in his thesis. However, we now
give an intermediate conjecture which Zauner does not state explicitly, but which can be
inferred from the fact that the group Hd exists for every d ≥ 2 together with the knowledge
of Zauner’s later stronger conjecture.
Conjecture 25. For each d ≥ 2, there is a maximal regular quantum design of degree 1,
r = 1, which is group covariant with respect to Hd .
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
36
Using the correspondence of equiangular lines and maximal regular quantum designs of
degree 1, r = 1, (Figures 1.1 and 1.2) we can restate Conjecture 25 as follows:
Conjecture 26. For each d ≥ 2, there is a set of d2 equiangular lines in Cd which is group
covariant with respect to Hd .
Example 27. Consider the lines corresponding to design D from Example 24. We have
seen that these vectors are group covariant with respect to G given in the same example.
Notice that the vectors satisfy the inner product condition of Proposition 9, and in fact, we
can extend this set to the following set of 9 = 32 equiangular lines in C3 which are group
covariant with respect to H3 :
x1 =
x2 =
x3 =
x4 =
x5 =
x6 =
x7 =
x8 =
x9 =
1
√ (1, ω, 0)T ,
2
1
√ (1, 0, ω)T ,
2
1
√ (0, ω, ω)T ,
2
1
√ (1, 1, 0)T ,
2
1
√ (1, 0, 1)T ,
2
1
√ (0, ω 2 , 1)T ,
2
1
√ (1, ω 2 , 0)T ,
2
1
√ (1, 0, ω 2 )T ,
2
1
√ (0, 1, ω 2 )T .
2
Here we note that X, Z described in §2.4 are actually the generators V, U of H2 . Thus
the construction of Hoggar’s 64 lines in C8 given by Godsil and Roy [23] is a variation on
the construction suggested by Conjecture 26, where instead of being group covariant with
respect to H8 , Hoggar’s lines are group covariant with respect to the 3-fold tensor product
of H2 . Recall that φ(H2 ) is called the Pauli group, and for this reason, the group Hd is
sometimes referred to as the generalized Pauli group.
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
2.5.2
37
Fiducial Vector Construction (A Second Refinement of Zauner’s
Conjecture)
To generate a design of the form (2.43), as suggested by Conjecture 25, one still needs an
initial projection matrix P1 of the design. Zauner [42, p. 59] also describes how one might
find such a matrix (see Figure 1.2).
Define a d × d matrix Zu = (zjk ), for j, k ∈ {0, . . . , d − 1}, by
zjk =
eπi(d−1)/12 πi(2jk+(d+1)k2 )/d
√
e
.
d
(2.44)
Then Zu is a unitary matrix (referred to by many authors as Zauner’s unitary) which
satisfies
Zu3 = Id
and acts on the Weyl matrices as follows:
Zu U
= V Zu
U Zu = eπi(d−1)/d Zu V −1 U −1
Zu V
=
eπi(d+1)/d U −1 V −1 Z
(2.45)
u.
Since U, V generate Hd , (2.45) tells us that
for each h ∈ Hd ,
hZu = αh Zu h0
for some αh ∈ S and some h0 ∈ Hd ,
(2.46)
for each h0 ∈ Hd ,
Zu h0 = βh0 hZu
for some βh0 ∈ S and some h ∈ Hd .
(2.47)
and
Notice that (2.46) and (2.47) imply that we can also write
for each h ∈ φ(Hd ),
hZu = αh Zu h0
for some αh ∈ S and some h0 ∈ φ(Hd ),
(2.48)
for each h0 ∈ φ(Hd ),
Zu h0 = βh0 hZu
for some βh0 ∈ S and some h ∈ φ(Hd ),
(2.49)
and
since any power of ω = e2πi/d can be absorbed into the constants αh and βh0 .
38
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Thus if we act on a Weyl-Heisenberg covariant design D by conjugation by Zu we obtain
another Weyl-Heisenberg covariant design. To see this, suppose D = {P1 , . . . , Pm } is Weyl-
Heisenberg covariant. Now consider Zu DZu−1 = {Zu P1 Zu−1 , . . . , Zu Pm Zu−1 }. To see that
Zu DZu−1 satisfies (i) of (2.41), notice that for h ∈ Hd ,
h(Zu Pj Zu−1 )h−1 = αh Zu h0 Pj h0−1 Zu−1 αh−1
= Zu (h0 Pj h0−1 )Zu−1
for some αh ∈ S and h0 ∈ Hd by (2.46)
and by Weyl-Heisenberg covariance of D, (h0 Pj h0−1 ) ∈ D which implies Zu (h0 Pj h0−1 )Zu−1 ∈
Zu DZu−1 . To see that (ii) of (2.41) is satisfied, by Weyl-Heisenberg covariance of D let
h0 ∈ Hd be such that Pj = h0 Pk h0−1 . Then
Zu Pj Zu−1 = Zu h0 Pk h0−1 Zu−1
= βh0 h(Zu Pk Zu−1 )h−1 βh−1
for some βh0 ∈ S and h ∈ Hd by (2.47)
0
= h(Zu Pk Zu−1 )h−1 .
Furthermore, if D has the form {AP1 A−1 : A ∈ Hd }, then it follows from (2.46) and
(2.47) that Zu DZu−1 has the form {AZu P1 Zu−1 A−1 : A ∈ Hd }.
Now let x be an eigenvector of Zu with eigenvalue 1 and let P = xx† . Take P to be our
initial projection matrix (x is called the fiducial vector associated to the initial projection).
Then Zauner’s construction forms a maximal quantum design as follows:
D = {V j U k P U −k V −j : j, k ∈ Zd }.
(2.50)
Note that whether this design is regular degree 1 with r = 1 is dependent on choosing an
appropriate eigenvector x.
This design is group covariant with respect to the Weyl-Heisenberg group as it has the
form (2.43). Furthermore, we have already seen that acting on D by conjugation by Zu gives
another Weyl-Heisenberg covariant design, where replacing Hd by φ(Hd ) as in (2.43) gives
Zu DZu−1 = {AZu P Zu−1 A−1 : A ∈ φ(Hd )}. Since P = xx† , Zu x = x, and Zu is unitary, this
means Zu DZu−1 = {AZu P Zu−1 A−1 : A ∈ φ(Hd )} = {AP A−1 : A ∈ φ(Hd )} = D, so that the
matrix Zu is also an automorphism (of order 3) of the design D (see Figure 1.2).
Zauner constructs examples of maximal regular quantum designs of degree 1, r = 1 of
the form (2.50) analytically for d = 2, 3, 4, 5; he constructs numerical examples (error within
10−6 ) for d = 6, 7.
With these examples in hand, Zauner made the following bold conjecture:
39
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Conjecture 28 ([42, p. 65]). For all d ≥ 2, there exist vectors, {x}, in the eigenspace
belonging to the eigenvalue 1 of the d × d matrix Zu , such that it is possible to generate
maximal regular degree 1, r = 1, quantum designs of the form
D = {V j U k P U −k V −j : j, k ∈ Zd },
starting from the d × d projection matrix P = xx† .
(We note that Zauner also shows that the eigenspace belonging to eigenvalue 1 has
dimension b d3 c + 1.)
Example 29. For d = 3, let ω = e2πi/3 . Then Zauner’s

1 ω2 ω2

iω
Zu = − √ 
1 1 ω
3
1 ω 1
matrix Zu is




which has eigenvectors x1 = (−ω, 0, 1)T , x2 = (−ω, 1, 0)T corresponding to eigenvalue 1. Let
x=
√1 (x1
2
− x2 ) =
√1 (0, −1, 1)T .
2
Then Zauner’s construction gives 9 projection matrices
{V a U b xx† U −b V −a : a, b ∈ Z3 }
 



0
0
0
0

1 
 1



=
0 1 −1  ,  0

2
2


0
0 −1 1

 
1 −1 0
1
 
1
 −1 1 0  , 1  −ω
 2
2
0
0 0
0

 
1 0 −1
1
 1
1
 0 0 0 ,  0
 2
2
−1 0 1
−ω 2
One can check for example

0 0
1
 0 1
2
0 −1
that
0


1
1

−1 
 2  −1
1
0
0
0


0
0
0
 1

−ω 2 
, 2  0
−ω
1
0
 
−ω 2 0
1
 1


1
0  ,  −ω 2
2
0
0
0


0 −ω
1
 1

0 0 
, 2  0
0 1
−ω
1
−1 0
1
0


0
0


−ω 
,
−ω 2 1

−ω 0

1 0 
,
0 0

0 −ω 2 


0
0 
 .


0
1
1
0

 1

 −1 1 0  ,
=
0 
 4

0
1 −1 0
which has trace 14 , and in fact for any distinct two of the projection matrices Pj , Pk , we have
tr(Pj Pk ) = 14 .
40
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
Notice that since U, V are unitary matrices, U −1 = U † , V −1 = V † . Thus an element of
a maximal regular degree 1, r = 1 quantum design constructed as in (2.43), as suggested
by Zauner, has the form V j U k xx† (U † )k (V † )j = (V j U k x)(V j U k x)† . This is simply the
projection matrix associated to the vector V j U k x, so we can equivalently consider the
elements of the design to be vectors of the form V j U k x; that is, Zauner’s construction builds
a Weyl-Heisenberg group covariant maximal regular degree 1, r = 1, quantum design by the
action of the Weyl matrices on a fiducial vector x, then takes the orthogonal projections
associated to the resulting vectors.
With this in mind, we can now see that a maximal regular Weyl-Heisenberg covariant
quantum design of degree 1, r = 1, generated by an eigenvector x of Zauner’s unitary with
eigenvalue 1, which has the form D = {Axx† A−1 : A ∈ φ(Hd )} = {Axx† A† : A ∈ φ(Hd )},
corresponds to a set of d2 Weyl-Heisenberg covariant equiangular lines in Cd of the form
L = {Ax : A ∈ φ(Hd )}.
(2.51)
We therefore give an equivalent form of Conjecture 28 phrased in terms of equiangular lines;
the following conjecture makes the statement of Conjecture 3 more precise:
Conjecture 30. For each d ≥ 2, there exists fiducial vectors, {x}, in the eigenspace belonging to the eigenvalue 1 of the d × d matrix Zu , such that
L = {V j U k x : j, k ∈ Zd }
is a set of d2 equiangular lines in Cd .
(Notice that in Figure 1.2, we write L = {αA Ax1 : A ∈ Hd }, which is the same as the
form given above if we take αA = ω −` for A = ω ` V j U k ∈ Hd .)
Example 31. We now give Example 29 in terms of equiangular lines. Start with eigenvector
x=
√1 (x1
2
− x2 ) =
a set of 9 vectors
√1 (0, −1, 1)T .
2
{V a U b x : a, b ∈ Z3 } =
Then Zauner’s construction for equiangular lines gives
1
1
1
√ (0, −1, 1)T , √ (0, −ω, ω 2 )T , √ (0, −ω 2 , ω)T ,
2
2
2
1
1
1
√ (−1, 1, 0)T , √ (−ω, ω 2 , 0)T , √ (−ω 2 , ω, 0)T ,
2
2
2
1
1
1
T
2
T
2 T
√ (1, 0, −1) , √ (ω , 0, −ω) , √ (ω, 0, −ω )
.
2
2
2
41
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
One can check that
1
1
√ (0, −1, 1)T , √ (−1, 1, 0)T
2
2
which has squared magnitude
1
4
=−
1
2
and in fact for any two distinct vectors xj , xk from this set,
we have |hxj , xk i|2 = 41 , making this set equiangular by Proposition 9.
We emphasize that Conjecture 28 and Conjecture 30 do not state that every eigenvector
of Zu having eigenvector 1 will produce the desired design or set of equiangular lines. In fact,
we will see in §2.6 that considerable analysis of automorphism groups, as well as substantial
computing resources have been dedicated to finding the particular linear combinations of
eigenvectors that can perform as fiducial vectors.
We extend the correspondence between designs and lines suggested by (2.51) by defining
an automorphism of a set of vectors (representing a set of lines) L = {xj } so that it is
analogous to an automorphism of a quantum design (see Figure 1.2). Let A be a unitary
matrix, then A is an automorphism of L if for each j there exists an αj ∈ S such that
xj = αj Axπ(j) ,
(2.52)
for some permutation π. Some authors refer to L as being left invariant under the action
of A. The fact that A induces a permutation π means that A defines a bijection from L to
itself. In particular we could equivalently write that
xj = αj Axk
for some k,
(2.53)
since if A maps two elements of L, say xk , x` to the same xj , we have
αj Axk = βj Ax`
which implies αj xk = βj x` since A is invertible; since xk , x` are elements of a set of
equiangular lines, we then have αj = βj and xk = x` , (otherwise xk , x` would differ only
by a phase, which by appropriate normalization, makes their inner product of magnitude 1
instead of the required magnitude
√ 1 ).
d+1
Notice if we form the corresponding projection matrices, we have
xj x†j = (αj Axπ(j) )(αj Axπ(j) )†
= (αj Axπ(j) )(x†π(j) A† αj )
= A(xπ(j) x†π(j) )A−1 ,
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
42
since A is unitary and αj ∈ S, and this is exactly the definition of automorphism of a design
from (2.36).
Conversely, if A is an automorphism of the design D = {xj x†j } corresponding to L =
{xj }, then Axπ(j) x†π(j) A−1 = xj x†j for some permutation π. Since A is unitary we have
Axπ(j) x†π(j) A−1 = Axπ(j) (Axπ(j) )† = xj x†j ⇔ αj Axπ(j) = xj
for some αj ∈ S, which means A satisfies the definition of automorphism for vectors from
(2.52).
Recall for quantum designs, we saw that having a transitive subgroup of the automorphism group of a design meant that we could generate the design from this subgroup and
an initial projection. We then saw that this was equivalent to group covariance by Proposition 22. Since we have now defined automorphisms for vectors and we have Proposition 23,
we can see that an analogous equivalence holds for a transitive subgroup of the automorphism group of a set of lines.
Now if as in Conjecture 30 one takes a fiducial vector x which is an eigenvector of Zu
having eigenvalue 1, then the set of vectors generated by the action of φ(Hd ) on x has Zu
as an automorphism, as noted by Zauner, since for each h ∈ φ(Hd ) there exists αh ∈ S such
that
hx = hZu x
x is an eigenvector of Zu with eigenvalue 1
= αh Zu h0 x
for some αh ∈ S and h0 ∈ φ(Hd ) by (2.48)
so that (2.52) is satisfied. Similar to our justification for (2.53), it is not hard to check that
Zu defines a bijection from φ(Hd ) to itself.
2.6
Numerical Solutions
Even though exact solutions have so far have been found only in dimensions 2–16, 19, 24,
28, 35 and 48 (see Table 1.1), numerical solutions in several other dimensions suggest that
Conjecture 2 may still hold. It is also hoped that numerical solutions will shed light on the
structure of possible new fiducial vectors.
There are two papers which make major contributions to our knowledge of the existence
of numerical maximum-sized sets of equiangular lines. These papers investigate maximumsized sets of equiangular lines as SIC-POVMs (symmetric informationally-complete positive
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
43
operator-valued measures), where a SIC-POVM is a set of d2 unit vectors {xj } in Cd such
that
|hxj , xk i|2 =
1
,
d+1
for all j 6= k.
(2.54)
Clearly, this is exactly the equiangularity condition of Proposition 9. However, in quantum
theory, SIC-POVMs are also often viewed as the “subnormalized” operators { d1 xj x†j }, which,
if we ignore the subnormalization, gives us a set of d2 orthogonal projections {xj x†j }. This
makes a SIC-POVM also equivalent to Zauner’s maximal regular degree 1, r = 1, quantum
design. We will represent a SIC-POVM as a set of vectors to highlight the equivalence with
complex equiangular lines (see Figure 1.1).
In 2004, a paper by Renes, Blume-Kohout, Scott and Caves [38] studying equiangular
lines as SIC-POVMs sparked more intense interest in the search for maximum-sized sets of
equiangular lines, particularly in the quantum information community. The authors only
construct exact SIC-POVMs in dimensions 2,3 and 4; however, they construct numerical
solutions (with error less than 10−8 ) in all dimensions up to 45, which support both their
own conjecture (as described below) and Zauner’s conjecture.
The paper [38] concentrates mainly on group covariant SIC-POVMs, in particular, with
respect to the representation of the group Zd × Zd by φ(Hd ), as described in §2.5. In fact,
the authors conjecture the following:
Conjecture 32 ([38, Conjecture 1]). For each d ≥ 2, let {bk }d−1
k=0 be an orthonormal basis
for Cd , and define
ω=e
2πi/d
,
Djk = ω
jk/2
d−1
X
ω jm bk⊕m b†m ,
m=0
where ⊕ denotes addition modulo d. Then there exists a unit vector x ∈ Cd such that the
set {Djk x}dj,k=1 is a SIC-POVM.
One can check that when one takes {bk }d−1
k=0 to be the standard basis, Conjecture 32
actually concerns Weyl-Heisenberg covariant SIC-POVMs of the form (2.51) by noticing
that, modulo phases, we then have Djk = W (d − k, j). To see this notice that the action
d−1
of Weyl matrices V and U on the standard basis {bj }j=0
for Cd implies that they can be
written as follows:
U
=
d−1
X
m=0
ω m bm b†m
44
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
V
=
d−1
X
bm b†m⊕1 .
m=0
Thus we can write
W (d − k, j) = V d−k U j
=
=
=
=
d−1
X
b` b†`⊕(d−k)
`=0
d−1
d−1
XX
`=0 m=0
d−1
d−1
XX
d−1
X
ω
jm
bm b†m
m=0
!
b` b†`⊕(d−k) ω jm bm b†m
ω jm b` hb`⊕(d−k) , bm ib†m
`=0 m=0
d−1
X
jm
ω
!
bk⊕m b†m
m=0
by (2.2)
since {bm } is an orthonormal basis
= ω −jk/2 Djk .
Example 33. A fiducial vector for d = 3 given in [38] is x =
√1 (0, −1, 1)T
2
(which is the
same fiducial vector given in Example 31). Then Renes et al. give the following set of lines:
1
1
1
√ (0, −1, 1)T , √ (0, −ω, ω 2 )T , √ (0, −ω 2 , ω)T ,
{Djk x : j, k ∈ {1, 2, 3}} =
2
2
2
1
1
1
√ (−1, 1, 0)T , √ (−ω 2 , 1, 0)T , √ (−ω, 1, 0)T ,
2
2
2
1
1
1
T
5/2
3/2 T
2
T
√ (1, 0, −1) , √ (ω , 0, −ω ) , √ (ω , 0, −1) , .
2
2
2
It is not hard to see that these are the same lines as given by Zauner’s construction in
Example 31 (up to phases).
Notice that Conjecture 32 is essentially Conjecture 25, in that it posits the fiducial vector
construction method using the group Hd will always give equiangular lines; however, unlike
Conjecture 28 it does not make any claim as to the origin of the fiducial vector. Since the
authors of [38] were not aware of the work done by Zauner in his thesis [42], their numerical
results are only shown to support their own Conjecture 32, and as a result Conjecture 25.
However, despite the fact that the authors were unaware of Conjecture 28, their numerical
fiducial vectors were later shown by Appleby [1] to also support that conjecture.
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
45
To obtain their numerical results, Renes et al. exploit a connection to frame theory.
They show that every POVM whose elements correspond to vectors in Cd is also a tight
frame; that is for {xk } a collection of vectors corresponding to POVM elements, there exists
a constant 0 < a < ∞ such that
a|hy, yi|2 =
X
k
|hy, xk i|2
(2.55)
for all y ∈ Cd . Furthermore, they show that a SIC-POVM is a minimal spherical 2-design
and that such designs minimize a quantity known as the second frame potential. They then
search for fiducial vectors which generate a set of d2 normalized vectors (via Zd × Zd ) such
that the second frame potential of the vectors is minimized.
In 2010, Scott and Grassl perform an extensive computer study of Weyl-Heisenberg
covariant SIC-POVMs. Once again, they use a connection between SIC-POVMs and frame
theory to compute numerical SIC-POVMs (accurate to 38 decimal places) for dimensions
≤ 67. They determine not only that Weyl-Heisenberg covariant SIC-POVMs exist in these
dimensions, but that in every dimension (except 66) there is a fiducial vector satisfying
Conjecture 28. Their search is exhaustive in dimensions ≤ 50, giving the number of fiducial
vectors (that generate unique orbits under the action of the Clifford group (see §2.7) in each
dimension).
The authors also give new exact solutions for d = 14, 24, 35, 48. In the case of d =
24, 35, 48, the authors were inspired by their numerical solutions, which revealed that the
automorphism groups of SIC-POVMs in these dimensions are comparatively large. Such
large automorphism groups allow for significant restriction of the search space of fiducial
vectors, making the computation of exact solutions feasible.
2.7
Clifford Group
We have just seen that the automorphisms of SIC-POVMs are important to the search
for numerical fiducial vectors. Recall from §2.5.1 that in studying quantum designs, and
in particular group covariant designs, the automorphisms of these designs are of interest.
Since SIC-POVMs are equivalent to certain designs (Figure 1.1), this is further incentive
to study their automorphisms. Furthermore, in the case of group covariant SIC-POVMs
having the form {Ax : A ∈ G}, there are only d complex variables (the components of the
fiducial vector x) that determine the whole SIC-POVM. With appropriate normalization of
46
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
x, this gives 2d − 1 real variables which must satisfy the
d
2
non-equivalent equations given
by (2.54); the existence of such SIC-POVMs in spite of this significant over-determination of
the fiducial vector suggests to researchers that SIC-POVMs might have additional concealed
symmetry which makes the study of their automorphisms of particular importance [3]. Since
the majority of known SIC-POVMs are group covariant with respect to Hd (that is, they
have Hd as a transitive subgroup of their automorphism group and φ(Hd ) as a regular
subgroup), many researchers study the normalizer of Hd in the group of unitary matrices.
The normalizer is given by
Cd = {complex d × d unitary A : AHd A−1 = Hd }
(2.56)
and is known as the Clifford group, or (a close relative of) the Jacobi group. We shall
show in Proposition 35 if we construct a SIC-POVM P as suggested in Conjecture 32 (and
Conjecture 26 using Figure 1.1) by
P = {hx : h ∈ φ(Hd )}
for some fiducial vector x, then Ax is also fiducial for every A ∈ Cd (see Figure 1.2). Before
we can see this, we first require the following lemma:
Lemma 34. For A ∈ Cd , we have
Aφ(Hd )A−1 = φ(Hd ).
(That is, conjugation by A permutes the cosets of Z(Hd ) in Hd .)
Proof. First, recall Z(Hd ) = {Id , ωId . . . , ω d−1 Id }, so that the center of Hd commutes with
any element of Cd .
Now we show that Aφ(Hd )A−1 ⊆ φ(Hd ). For each AhZ(Hd )A−1 ∈ Aφ(Hd )A−1 , we have
AhZ(Hd )A−1 = AhA−1 Z(Hd )
= h0 Z(Hd )
for some h0 ∈ Hd since A ∈ Cd ,
which is in φ(Hd ), as required.
Next we show that Aφ(Hd )A−1 ⊇ φ(Hd ). For each h ∈ Hd , h = Ah0 A−1 for some
h0 ∈ Hd , since A ∈ Cd . Then for hZ(Hd ) ∈ φ(Hd ), we have
hZ(Hd ) = Ah0 A−1 Z(Hd )
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
47
= Ah0 Z(Hd )A−1
which is in Aφ(Hd )A−1 , as required.
Proposition 35 ([39]). If x is a fiducial vector such that P = {hx : h ∈ φ(Hd )} is a
SIC-POVM, then for each A ∈ Cd , Ax is also such a fiducial vector.
Proof. Let A ∈ Cd . By Lemma 34, we have
{hAx : h ∈ φ(Hd )} = {Ah0 x : h0 ∈ φ(Hd )}.
Now consider two distinct elements Ahx, Ah0 x of {Ah0 x : h0 ∈ φ(Hd )}. The squared
magnitude of their inner product is
|hAhx, Ah0 xi|2 = |(Ahx)† (Ah0 x)|2
by (2.2)
= |(x† h† A† )(Ah0 x)|2
= |x† h1 x|2
= |hx, h1 xi|2
1
=
d+1
for h1 = h† h0 = h−1 h0 ∈ φ(Hd ) and since A† = A−1
by (2.2)
by (2.54) since x, h1 x ∈ P and h1 6= Id .
Proposition 35 tells us that each fiducial vector x generates an orbit of fiducial vectors
{Ax : A ∈ Cd } under the action of the Clifford group, as noted in §2.6 (see Figure 1.2).
Many authors then count the number of inequivalent fiducial vectors, where they define
inequivalent as generating distinct orbits under the action of the Clifford group. Also note,
for A ∈ Cd , the SIC-POVM P = {hAx : h ∈ φ(Hd )} is group covariant with respect to the
Weyl-Heisenberg group, as it has the form (2.51). Furthermore, the action of the Clifford
group on a Weyl-Heisenberg covariant SIC-POVM maintains Weyl-Heisenberg covariance
(see Figure 1.2).
Recall from (2.45) that Zauner’s unitary Zu satisfies the following relations:
Zu U
= V Zu
U Zu = eπi(d−1)/d Zu V −1 U −1
Zu V
= eπi(d+1)/d U −1 V −1 Zu .
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
48
Now recall that elements of Hd have the form ω ` V j U k for `, j, k ∈ Zd and ω = e2πi/d .
Some authors allow Hd to contain elements of the form τ ` V j U k for j, k ∈ Zd , τ = eπi/d
and ` ∈ Z2d . Since this simply multiplies elements of Hd by additional phases, it enlarges
the center Z(Hd ) and leaves the group φ(Hd ) unchanged. Since we have already seen that
we can use φ(Hd ) in place of Hd when constructing Weyl-Heisenberg covariant designs and
vectors, it leaves our previous results unchanged to enlarge the group Hd in such a way.
However, with this definition of Hd , we can see that U, V, τ generate Hd , and from (2.45)
we get that Zu normalizes Hd so that Zu ∈ Cd . For the remainder of this section we take
Hd to be this enlarged group.
Now Conjecture 30 suggests that fiducial vectors should be found among the eigenvectors
of Zu . Since we have seen that Zu ∈ Cd , it is natural to ask whether there are other elements
of Cd whose eigenvectors may be used as fiducial vectors. Grassl [25] addresses exactly this
question by searching for a fiducial vector which is an eigenvector with eigenvalue 1 of
an element of the Clifford group. Assuming the existence of a Weyl-Heisenberg covariant
SIC-POVM generated by such a fiducial vector yields a system of polynomial equations
in at most 2d variables, which he solves using the computer algebra system Magma. In
this way he is able to give the first exact set of equiangular lines in a non-prime power
dimension, namely d = 6 [25]. An example of one such fiducial vector that he found is
x = θ3 (v1 , v2 , v3 , v4 , v5 , v6 )T , where
s √
3 21 + 9
,
θ1 =
224
θ2 is the real root of
s √
√
√
3 21 + 3 7 + 7 3 + 21
θ3 =
,
148384
√
√
21 − 3
21 − 5 21
x+
θ1 ,
x −
28
126
3
and the vj are given below.
√
√
√
√
√
v1 = ((336( 7 − 21)θ1 − 42 21 − 42 3 − 126 7 − 378)θ22
√
√
√
√
√
+(56(3 7 − 2 3 + 3)θ1 + 3 21 − 21 3 + 9 7 + 63)θ2
√
√
√
√
√
√
+(168 − 24 21 − 56 3 + 24 7)θ1 + 6 21 + 18 3 − 6 7 − 6)i
√
√
√
√
√
+(336( 7 + 21)θ1 + 42 21 − 42 3 − 126 7 + 378)θ22
√
√
√
√
√
+(56(3 7 − 2 3 − 3)θ1 − 3 21 − 21 3 + 9 7 − 63)θ2
√
√
√
√
√
√
+(24 21 − 56 3 + 24 7 − 168)θ1 − 6 21 + 18 3 − 6 7 + 6,
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
49
√
√
√
v2 = ((672( 7 − 21)θ1 − 168 3 + 504)θ22
√
√
√
√
+(28(3 21 + 5 3 − 3 7 − 15)θ1 − 42 3 + 126)θ2
√
√
√
√
√
√
+(336 − 48 21 − 112 3 + 48 7)θ1 − 12 21 − 12 3 + 12 7 + 36)i
√
√
√
−(84 21 − 252 3 − 252 7 + 252)θ22
√
√
√
√
√
√
+(84( 21 + 3 − 3 7 − 1)θ1 − 6 21 + 18 7)θ2 − 24 3 + 24,
√
√
√
√
√
v3 = 6( 7 − 3)i + 6 21 + 12 3 − 12 7 − 18
√
√
√
√
√
v4 = ((336( 7 − 21)θ1 + 126 21 − 42 3 − 126 7 + 126)θ22
√
√
√
√
√
√
+(56(6 − 3 21 − 2 3 + 3 7)θ1 − 9 21 − 21 3 + 9 7 + 63)θ2
√
√
√
√
√
√
+((168 − 24 21 − 56 3 + 24 7)θ1 + 6 21 + 18 3 − 6 7 − 54))i
√
√
√
√
√
+(336( 21 − 3 7)θ1 + 42 21 − 378 3 − 126 7 + 378)θ22
√
√
√
√
+(168( 3 − 1)θ1 − 3 21 + 63 3 + 9 7 − 63)θ2
√
√
√
√
√
√
+(24 21 + 168 3 − 72 7 − 168)θ1 + 6 − 6 21 − 6 3 + 18 7,
√
√
√
v5 = ((672 7θ1 + 84 21 − 168 3 + 252)θ22
√
√
√
√
√
−((84 21 − 140 3 + 84 7 − 84)θ1 − 6 21 + 42 3)θ2
√
√
√
√
−(112 3θ1 − 48 7θ1 + 12 3 − 12 7 + 24))i
√
√
√
+(672 7θ1 − 84 21 − 168 3 − 252)θ22
√
√
√
√
√
+((84 21 + 140 3 − 84 7 − 84)θ1 − 6 21 − 42 3)θ2
√
√
√
√
−112 3θ1 + 48 7θ1 − 12 3 + 12 7 + 24,
√
√
√
v6 = 6( 7 − 3)i − 6 21 + 18.
Further exploiting the importance of the Clifford group to the construction of SICPOVMs, Appleby, Bengtsson, Brierley, Grassl, Gross and Larsson [4] give simplified fiducial
vectors for d = 4 and 9 and give the first exact fiducial vectors for d = 16. Their major
insight is a new representation of Hd , which leads to a simplified representation of Cd ,
allowing for easier computation of fiducial vectors.
In its standard representation, given in §2.5, the Weyl-Heisenberg group is generated by
a diagonal unitary matrix U and a permutation matrix V . As a result, this representation
consists of phase-permutation matrices; that is, unitary matrices having only one nonzero
entry in each row and in each column. With this representation, it is not the case that
the Clifford group can be represented by phase-permutation matrices (look, for example, at
50
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
the form of Zauner’s unitary given in (2.44)). However, in square dimensions d = n2 , the
authors of [4] give a new representation of Hd , such that the Clifford group consists only of
phase-permutation matrices.
This representation replaces matrices V, U by matrices X, Z such that X n , Z n are diagd
onal. This diagonalization defines a new basis {b(j,k) }n−1
j,k=0 for C , where

b
if k + 1 6= 0 mod n,
(j,k+1)
Xb(j,k) =
(e2πi/n )j b
if k + 1 = 0 mod n
(j,0)
Zb(j,k) = (e2πi/d )k b(j−1,k) .
Example 36. Let d = 4. Then n = 2 and the basis elements are labelled by (0, 0), (0, 1),
(1, 0), and (1, 1). Then X and Z are given below, where the row and column labels indicate
basis elements
(0, 0)
and

(0, 0) (0, 1) (1, 0) (1, 1)
0

X = (0, 1) 
 1

(1, 0)  0

(1, 1)
0
(0, 0)

1
0
0
0
0
0
0
1
0

0 


−1 

0
(0, 0) (0, 1) (1, 0) (1, 1)
0

Z = (0, 1) 
 0

(1, 0)  1

(1, 1)
0

0
1
0
0
0
i
0
0
i
0





0 

0
Using this new basis, the Clifford group is now represented by phase-permutation matrices. This greatly simplifies the computation of fiducial vectors. The computation for
d = 9 can now be done by hand, in contrast to the solution given in [39] which was done
using Magma (§A.2). It also allows for the first exact solution for d = 16 to be computed,
using Gröbner bases and modular techniques to solve polynomial equations resulting from
the equiangularity condition (2.54) (see [28] for details of the method used).
Even with this simplified representation of Cd , the authors of [4] needed 3 days and
30 GB of memory to compute a Gröbner basis modulo a single 23-bit prime using Magma.
The coefficients of the resulting polynomials in the basis have nearly 900 digits, and 3 pages
of [4] are dedicated to listing the fiducial vector components for d = 16 (§A.4).
CHAPTER 2. BACKGROUND AND PREVIOUS WORK
51
The ideas of [4] are extended by Appleby, Bengtsson, Brierley, Ericsson, Grassl and
Larsson in [3]. Instead of restricting to square dimensions, where the Clifford group has a
phase-permutation (also known as a monomial ) representation, the authors generalize to
arbitrary dimension d = kn2 , where k is square-free. In this case they show that the Clifford
group can be represented by k-nomial matrices; that is, matrices which can be formed by
taking an n2 × n2 monomial matrix M and replacing each entry of the matrix by a k × k
block, where a k × k block is nonzero if and only if it corresponds to a nonzero entry of M .
The resulting representation of the Clifford group once again facilitates the calculation
of exact fiducial vectors in certain dimensions. The authors are able to calculate fiducial
vectors for d = 8 by hand. For d = 12, the authors achieve a speed-up of three orders
of magnitude over the same calculation using the standard representation and reduce the
amount of memory required from 17 GB to less than 100 MB. They also achieve a considerable simplification of the fiducial vector itself (§A.3); the components of the fiducial vector
from [3] require just under a page to write down, whereas the fiducial vector in [28], found
using the standard representation of the Clifford group, occupies 6 pages. Furthermore the
authors of [3] give the first exact fiducial vector for d = 28.
Even with the considerable simplification and speed-ups achieved in the calculation of
exact solutions satisfying Conjecture 28, the methods of [3, 4, 28, 39] are rapidly becoming infeasible as dimension increases. Even the simplified fiducial vectors are dauntingly
complicated (Appendix A); however, the solutions for dimensions 4 and 8 given in §2.3 and
§2.4 suggest that a construction which results in far simpler solutions might exist for other
dimensions as well.
Chapter 3
New Constructions
The possible computational improvements to the search for exact solutions satisfying Conjecture 28 seem to have been largely exhausted. Rather than directing further resources
toward this endeavour, our exploration, which we describe in this chapter, concentrates on
finding connections between equiangular lines and other combinatorial objects in order to
produce simpler exact solutions.
We will give three new constructions for maximum-sized sets of equiangular lines. In
§3.1, we see the first construction which builds sets of d2 equiangular lines in Cd from d MUBs
constructed from a (d, d, d, 1)-RDS. The solutions given by this construction were previously
known, but they illustrate a connection to MUBs which has not been observed before. In
§3.2, we see the second construction which builds maximum-sized sets of equiangular lines
from a single d × d complex Hadamard matrix. This relationship between equiangular lines
and Hadamard matrices has not been noticed before and it leads to a new solution for d = 8,
which is arguably the simplest known set of 64 lines in that dimension. In §3.3, we see the
third construction which produces another new simple maximum-sized set of equiangular
lines in C8 . This construction also links sets of equiangular lines in different dimensions as
it uses building blocks in a smaller dimension to construct lines in twice that dimension.
3.1
Construction from MUBs
The simplest maximum-sized set of equiangular lines that one might hope for would be one
in which all the lines are flat. However, as noted at the end of §2.2, the largest number of
such lines that is possible is d2 − d + 1 < d2 . It is then natural to ask whether one can
52
53
CHAPTER 3. NEW CONSTRUCTIONS
construct maximum-sized sets of equiangular lines which are almost flat, that is, lines in
which all but one entry of the vector have the same magnitude. This has been accomplished
by Zauner [42] and Renes et al. [38] for d = 3, by Belovs [9] and Appleby et al. [4] for d = 4
and by Appleby [1] for d = 7, 19. Here we give our first construction which also results in
maximum-sized sets of almost flat equiangular lines.
Our construction begins with a set of d MUBs in Cd , formed from an RDS, as in Theorem 17 but with the standard basis excluded, and multiplies a single entry of each vector
by some constant. This construction creates sets of maximum-sized equiangular lines in dimensions 2, 3 and 4. These sets were already known from the fiducial vector construction of
§2.5.2 (in particular, they comprise some of the previously mentioned almost flat solutions);
however, our construction highlights a connection between equiangular lines and MUBs and
RDSs that has not previously been exploited in this way.
Let B1R , . . . , BdR be d MUBs in Cd formed from a (d, d, d, 1)-RDS R. Let π be a permutation of {1, . . . , d}. Let BjR (π, v) denote the block of vectors such that each vector in BjR
has been multiplied in coordinate π(j) by v. Let
LR
d (π, v)
=
d
[
BjR (π, v).
(3.1)
j=1
Example 37. Consider the following MUBs in C2 obtained from the (2, 2, 2, 1)-RDS R =
∼ Z2 , as in Theorem 17:
{1, x} in hxi = Z4 relative to hx2 i =
B1R =
(
(1
B1R (π, v) =
(
(v
1 )T
(1 −1)T
)
B2R =
(
i )T
(1
(1 −i)T
)
.
Let π = [1, 2]. Then
1 )T
(v −1)T
)
and
LR
2 (π, v) =
B2R (π, v) =


(v





 (v
(

1 )T 




T

−1 ) 


(1 iv )T




 (1 −iv)T







.
(1
iv )T
(1 −iv)T
)
,
54
CHAPTER 3. NEW CONSTRUCTIONS
Theorem 38. For d = 2, 3, 4 there exists a (d, d, d, 1)-RDS R, a permutation π ∈ Sd and
2
complex constant v(d) such that LR
d (π, v(d)) is a set of d almost flat equiangular lines in
Cd .
The proof of Theorem 38 is the following 3 examples:
Example 39. d = 2 : Take the (2, 2, 2, 1)-RDS R of Example 37. This gives the set of 2
MUBs of Example 37. Take the permutation π = [1, 2] and constant v = a + ib. Then our
construction gives the set of vectors
LR
2 (π, v) =
which are equiangular for a+ib ∈


(a + ib





 (a + ib


(




(
1
1
1
−1
i(a + ib)

)T 




T

) 
)T
−i(a + ib))T







,
√
√
√ 1 √
√
√
√ 1 √
1
1
2 ( 2 ± 6), − 2 ( 2 ± 6), 2 i( 2 ± 6), − 2 i( 2 ± 6) .
Example 40. d = 3 : Take the (3, 3, 3, 1)-RDS R = {1, y, xy 2 } in G = hxi × hyi = Z3 × Z3
∼ Z3 . The resulting MUBs are as follows:
relative to hxi × h1i =
(1
y
χ0,0 (1
1
χ0,1 (1
ω
χ0,2 (1 ω 2
χ1,0 (1
1
χ1,1 (1
ω
χ1,2 (1 ω 2
χ2,0 (1
1
χ2,1 (1
ω
χ2,2 (1 ω 2
xy 2 )T

1 )T 

ω 2 )T
B1R


ω )T

ω )T 

1 )T
B2R


ω 2 )T

ω 2 )T 

T
ω )
B3R


1 )T
where ω = e2πi/3 . Take the permutation π = [1, 2, 3] and v = a + ib. Then our construction
55
CHAPTER 3. NEW CONSTRUCTIONS
gives the set of vectors
LR
3 (π, v)
=


 (a + ib




(a + ib





(a + ib









( 1
(




(








(





(




(
1
1
ω
ω2
ω2
ω
a + ib
ω
1
(a + ib)ω
1
1
(a + ib)ω 2
1
1
1
ω
1
ω2

)T 




)T 




T
) 







T
) 


)T



ω2
)T 






2
T

(a + ib)ω ) 



T

(a + ib)ω ) 




T
a + ib )
which are equiangular for a = b = 0.
,
Example 41. d = 4 :Take the (4, 4, 4, 1)-RDS, R = {1, x, y, x3 y 3 } in G = hxi×hyi = Z4 ×Z4
relative to hx2 i × hy 2 i ∼
= Z2 × Z2 , as in Example 18. Take the permutation π = [1, 3, 4, 2]
and v = a + ib. Then our construction gives the


(a + ib
1





(a + ib
1





(a + ib
−1






(a + ib
−1








( 1
1





( 1
1





( 1
−1





( 1
−1
LR
(π,
v)
=
4


i
( 1





( 1
i





( 1
−i






( 1
−i







( 1
i(a + ib)






( 1
i(a + ib)





( 1
−i(a + ib)




( 1
−i(a + ib)
set of vectors
1
1
−1
−1
−1
1
1
i(a + ib)
−i(a + ib)
−1
−i
i
i(a + ib)
i
−i(a + ib)
−i
1
−1
−i(a +

)T 



T

) 




T

) 



T

) 






T

) 



T

) 




T
) 




T

) 
ib))T
i(a + ib) )T
1
i(a + ib) )T
−1
−i(a + ib))T
−1
)T
1
)T
i
1
)T
−i
−1
)T
i
−i









































,
CHAPTER 3. NEW CONSTRUCTIONS
which are equiangular for a + ib ∈
56
np
p
p
√
√ p
√
√ o
2 + 5, − 2 + 5, i 2 + 5, −i 2 + 5 .
Notice that in Examples 39, 40, and 41, there is at least one v which is real, and every
v is either entirely real or entirely imaginary; as a result it is easily verified that these lines
satisfy (2.5). The constants v and permutations π in the previous examples were found
for the given R by calculating all inner products between distinct lines for each π ∈ Sd
and solving the equations that result from (2.5) using the computer algebra system Maple.
However, one need only calculate the inner product of the first line with another line from
the first block and with each line in the second block to yield enough equations to determine
v, if there is a suitable v for a given π. It is then easy to check if the pair (v, π) yields a
set of equiangular lines. The equations that must be solved to determine v = a + ib end up
being quartic in a and b, and are simple enough to be solved by hand.
We will address the case d > 4 at the end of this section, but now we give a necessary
condition on the magnitudes of the entries in a set of almost flat equiangular lines of the form
LR
d (π, v). Khatirinejad [35] proves the following result about almost flat fiducial vectors for
the construction described in §2.5.2:
Theorem 42 ([35, Theorem 3.1]). Let z be a fiducial vector in Cd such that one coordinate
of z has absolute value m and all other coordinates have absolute value n. Then
√
√
1 ± (d − 1)/ d + 1
1 ∓ 1/ d + 1
2
2
,
m =
.
n =
d
d
Here we adapt Theorem 42 by removing the assumption that our set of equiangular lines
is constructed from a fiducial vector and adapting the result to apply to the construction of
equiangular lines from a set of d MUBs.
Form d MUBs in Cd from a (d, d, d, 1)-RDS R as in Theorem 17. Notice that for this
construction of MUBs, each vector of the bases B1R , . . . , BdR has all entries that are roots of
√
unity. Thus each vector has norm d, so from (2.24) we can see that for x, y ∈ ∪dj=1 BjR ,


d



|hx, yi| = 0



√
if x = y
if x 6= y but they are in the same basis
d
if x, y are in different bases.
(3.2)
57
CHAPTER 3. NEW CONSTRUCTIONS
Given a permutation π ∈ Sd and a constant v ∈ C, form LR
d (π, v) as in (3.1). Then (3.2)
implies for x, y ∈ LR
d (π, v) that


d − 1 + |v|2
|hx, yi| = 
|v|2 − 1
if x = y
(3.3)
if x 6= y but they are in the same basis.
(For (3.3), we could derive an expression for when x, y are in different bases, but it will not
be necessary.)
Now we normalize the elements of LR
d (π, v) so that they are unit vectors, and (3.3)
becomes
|hx, yi| =


1
if x = y
2

 ||v| −1|
d−1+|v|2
(3.4)
if x 6= y but they are in the same basis.
We can now give our adaptation of Theorem 42.
Theorem 43. Given a set of d2 equiangular lines in Cd such that each line has a single
entry of magnitude m and all the rest have magnitude n, and the lines are constructed as
LR
d (π, v) and normalized so that each vector has norm 1, then
√
√
1 ∓ 1/ d + 1
1 ± (d − 1)/ d + 1
2
2
n =
,
m =
.
d
d
Proof. Let x, y be distinct normalized lines of LR
d (π, v) originating from the same basis. By
Proposition 9, we must have |hx, yi| =
√1 .
d+1
1
√
d+1
So
=
Then (3.4) tells us that
2
|v | − 1
.
d − 1 + |v|2
1
|v|2 − 1
= ±√
2
d − 1 + |v|
d+1
√
2
⇒ (|v| − 1) d + 1 = ±(d − 1 + |v|2 )
√
√
⇒ ( d + 1 ∓ 1)|v|2 = ±(d − 1) + d + 1
√
±(d − 1) + d + 1
2
√
⇒
|v| =
d+1∓1
√
= 2 ± d + 1.
58
CHAPTER 3. NEW CONSTRUCTIONS
Notice that before normalization each line in LR
d (π, v) has a single entry of magnitude
|v| and the rest have magnitude 1. After normalization, there is a single entry of magnitude
m= √
|v|
d−1+|v|2
and the rest have magnitude n = √
n2 =
=
=
=
=
1
.
d−1+|v|2
Thus
1
d − 1 + |v|2
1
√
d−1+2± d+1
√
d+1∓ d+1
(d + 1)2 − (d + 1)
√
(d + 1)(1 ∓ 1/ d + 1)
d(d + 1)
√
1 ∓ 1/ d + 1
d
and
m2 = |v|2 n2
√
√
1 ∓ 1/ d + 1
= (2 ± d + 1)
d
√
√
1 ± d + 1 ∓ 2/ d + 1
=
d√
1 ± (d − 1)/ d + 1
=
.
d
We note that since 2 ±
√
d + 1 = |v|2 ≥ 0, that we must have |v|2 = 2 +
d ≥ 4. In this case we are left with
√
1 − 1/ d + 1
2
n =
d
√
d + 1 when
√
1 + (d − 1)/ d + 1
m =
.
d
2
Notice that if we normalize the lines given in Examples 39, 40, and 41 to be unit vectors,
then the magnitudes of their entries satisfy Theorem 43.
Example 44. For d = 4, the lines given in Example 41 have v with magnitude
p
√
Each line has norm 5 + 5, so normalizing to obtain unit vectors we find that
√
2+ 5
2
√
m =
5+ 5
p
√
2 + 5.
59
CHAPTER 3. NEW CONSTRUCTIONS
=
n2 =
=
3
√
1+
5
1
√
5+ 5
1
1
1− √
,
4
5
1
4
satisfying Theorem 43. Furthermore, if we look at the example from Belovs [9] given in §2.3
and rearrange the columns of the given matrix, it is easy to see that the vectors given by the
p
√
columns of the matrix are identical to this normalized set of lines LR
2 + 5).
4 ([1, 3, 4, 2],
We note that for d = 2, 3 with the RDS given in Examples 39, 40, respectively, one can
choose any permutation π ∈ Sd and there will be a constant v(d) such that Theorem 38
holds (in fact, it holds for the same v(d) as given in the examples). However, in the case
d = 4 with the RDS of Example 41, only 8 of the 24 permutations in S4 result in a set
of equiangular lines, (all 8 permutations result in the same v from Example 41). We now
look the case d = 4 in more detail. Let R be fixed but arbitrary. We shall give a sufficient
p
√
condition for π ∈ S4 to result in LR
4 (π, ± 2 + 5) being a set of equiangular lines.
When d = 4, Theorem 17 constructs blocks B1R , B2R , B3R , B4R with elements that are
the restriction of the characters of Z4 × Z4 to the (4, 4, 4, 1) RDS-R. Notice that since the
exponent of Z4 × Z4 is 4, all vector entries are in {1, −1, i, −i} and each vector has norm
2. Now consider the BjR to be 4 × 4 matrices, whose columns comprise 4 MUBs in C4 . Let
π ∈ S4 and arrange the BjR into one 4 × 16 matrix B R (π) according to π as in Figure 3.1a.
Since we know that B1R , B2R , B3R , B4R are mutually unbiased bases with basis elements of
norm 2, by (2.24) and (2.21) we have

4I4
(BjR )† BkR =
(2eiθs,t )4
s,t=1
if j = k
(3.5)
if j 6= k,
for some real constants θs,t which depend on j, k, s, t.
Then decompose each BπR−1 (j) as CπR−1 (j) +DπR−1 (j) , as in Figure 3.1b, where C R (π), DR (π)
have the same entries as B R (π) except where shaded entries have been zeroed out. Note
that the arrangement of the BjR in Figure 3.1a according to the permutation π −1 means
that if we were to replace each block BjR by BjR (π, v) then the constant v would fall along
the “diagonal” of B R (π); that is, v would multiply the nonzero entries of matrix DR (π) of
Figure 3.1b.
60
CHAPTER 3. NEW CONSTRUCTIONS
B R (π) =
BπR−1 (1)
BπR−1 (2)
BπR−1 (4)
BπR−1 (3)
(a) Arrangement of 4 MUBs into a single matrix.
C R (π) =
CπR−1 (1)
CπR−1 (2)
CπR−1 (3)
CπR−1 (4)
DπR−1 (1)
DπR−1 (2)
DπR−1 (3)
DπR−1 (4)
DR (π) =
(b) Decomposition of B R (π) into 2 matrices C R (π), DR (π).
Figure 3.1
Example 45. Let B1R , B2R , B3R , B4R be defined as in Example 18 from RDS R = {1, x, y, x3 y 3 }
in G = hxi × hyi ∼
= Z4 × Z4 . Take π = [1, 3, 4, 2], then B R (π), C R (π), DR (π) are formed as
follows:
B R (π) =

BπR−1 (1)
1 1

 1 1

= 
 1 −1

1 −1
1
BπR−1 (2)
1
−1 −1
1 −1
−1 1
1
BπR−1 (3)
1
1
1
i
i −i −i
i −i i −i
−1 1
1 −1
1
BπR−1 (4)
1
1
1
1 −1 −1
i −i i
−i i
i
1
−i
−i
1
1
1
1


i −i −i 


1 −1 1 −1 

−i i i −i
i
61
CHAPTER 3. NEW CONSTRUCTIONS

0 0
0
0

 1 1 −1 −1

= 
 1 −1 1 −1

1 −1 −1 1

1 1 1 1

 0 0 0 0

+ 
 0 0 0 0

0 0 0 0
=
CπR−1 (1)
+
DπR−1 (1)
1
1 1 1
1 1 1
1
0
0 0 0
i
−i i −i
1 1 −1 −1
0 0 0
0
−1 1 1 −1
−i i
0
0
0
0
0
0
i
0
0
0
0
0
i −i −i
0
0
0
0
0
0
0
0
0
0
−i
i
i −i i −i
0
0
0
1 1
1
1
0
0


−i −i 


1 −1 1 −1 

i
i
0 0
0
0
0
0
0
0
0
0
−i i


0 


0 0 

i −i
CπR−1 (2)
CπR−1 (3)
CπR−1 (4)
DπR−1 (2)
DπR−1 (3)
DπR−1 (4)
= C R (π) + DR (π)
0
Consider the matrix B R (π, v) = C R (π) + vDR (π). Let BπR−1 (j) (v) = CπR−1 (j) + vDπR−1 (j) .
Then we already noted that {BπR−1 (j) (v) : 1 ≤ j ≤ 4} = {BjR (π, v) : 1 ≤ j ≤ 4} so from
R
(3.1) the columns of B R (π, v) are the vectors of LR
4 (π, v). For L4 (π, v) to be a set of 16
equiangular lines, we must consider the inner products of every pair of distinct vectors,
which using (2.21), we can now see are given by the off-diagonal elements of the matrix
(B R (π, v))† B R (π, v). The following theorem uses this viewpoint of LR
4 (π, v) as the columns
of B R (π, v) to characterize which permutations π result in a set of equiangular lines when
p
√
v = ± 2 + 5.
Theorem 46. Fix a (4, 4, 4, 1)-RDS R in Z4 × Z4 . For C R (π), DR (π) formed as above, the
p
√
columns of the matrix C R (π) ± DR (π) 2 + 5 comprise 16 equiangular lines in C4 if π is
√
such that all entries of (C R (π))† C R (π) have magnitude 2.
Proof. Let B R (π, v) = C R (π) + vDR (π) for real v. Let j 0 = π −1 (j) for each j = 1, . . . , 4.
Then BjR0 (v) = CjR0 + vDjR0 . We shall show that
(1) all off-diagonal entries of (BjR0 (v))† BjR0 (v) have magnitude |v 2 − 1|, and
(2) all entries of (BjR0 (v))† BkR0 (v) for j 0 6= k 0 have magnitude
√
2v 2 + 2.
Therefore, the columns of B R (π, v) form 16 equiangular lines in C4 exactly when v satisfies
|v 2 − 1| =
p
2v 2 + 2,
62
CHAPTER 3. NEW CONSTRUCTIONS
p
√
which is solved by v = ± 2 + 5 to give the result.
We have
(CjR0 )† + v(DjR0 )† (CkR0 + vDkR0 )
= (CjR0 )† CkR0 + v (CjR0 )† DkR0 + (DjR0 )† CkR0 + v 2 (DjR0 )† DkR0 .
(BjR0 (v))† BkR0 (v) =
(3.6)
To prove (1), set j 0 = k 0 in (3.6) to give
(BjR0 (v))† BjR0 (v) = (CjR0 )† CjR0 + v (CjR0 )† DjR0 + (DjR0 )† CjR0 + v 2 (DjR0 )† DjR0
= (CjR0 )† CjR0 + v 2 (DjR0 )† DjR0 ,
(3.7)
since (CjR0 )† DjR0 = (DjR0 )† CjR0 = 0 because of the zeroed-out elements indicated by Figure 3.1b.
Now since BjR0 (1) = BjR0 , by (3.5) we have
4I4 = (BjR0 )† BjR0
= (CjR0 )† CjR0 + (DjR0 )† DjR0
by (3.7) with v = 1.
(3.8)
Substituting the expression for (CjR0 )† CjR0 from (3.8) into (3.7) we get
(BjR0 (v))† BjR0 (v) = 4I4 + (v 2 − 1)(DjR0 )† DjR0 .
Since all entries of (DjR0 )† DjR0 have magnitude 1, this implies all the off-diagonal entries of
(BjR0 (v))† BjR0 (v) have magnitude |v 2 − 1|.
To prove (2), take j 6= k in (3.6) to get
(BjR0 (v))† BkR0 (v) = (CjR0 )† CkR0 + v (CjR0 )† DkR0 + (DjR0 )† CkR0 + v 2 (DjR0 )† DkR0
= (CjR0 )† CkR0 + v (CjR0 )† DkR0 + (DjR0 )† CkR0
(3.9)
since (DjR0 )† DkR0 = 0 for j 6= k because of the zeroed-out elements of Figure 3.1b. Now put
v = 1 in (3.9) to give
(BjR0 )† BkR0 = (CjR0 )† CkR0 + (CjR0 )† DkR0 + (DjR0 )† CkR0 .
(3.10)
Fix some position in the matrix (BjR0 )† BkR0 . By assumption, the entry of (CjR0 )† CkR0 at this
√
position has magnitude 2 and is a sum of elements from {1, −1, i, −1}; thus it has the
form a + ib for a, b ∈ {1, −1}. Since the columns of BkR0 and the columns of BjR0 form
unbiased bases with {1, −1, i, −1} entries, by (3.5) the entry of (BjR0 )† BkR0 in this position
CHAPTER 3. NEW CONSTRUCTIONS
63
has magnitude 2 and is also a sum of elements of {1, −1, i, −1}. By (3.10), this forces the
entry of (CjR0 )† DkR0 + (DjR0 )† CkR0 in the same position to be a − ib or −a + ib. By (3.9), the
entry of (BjR0 (v))† BkR0 (v) in this position is either a + ib + v(a − ib) or a + ib + v(−a + ib),
p
√
which in both cases has magnitude (1 + v)2 + (1 − v)2 = 2v 2 + 2, as claimed.
If we check the condition of Theorem 46 on (C R (π))† C R (π) for the RDS R given in
Example 41 over all permutations π ∈ S4 , we find that there are only 8 for which the as-
sumption holds, and they are [1, 3, 4, 2], [1, 4, 2, 3], [2, 4, 3, 1], [2, 3, 1, 4], [3, 2, 4, 1], [3, 1, 2, 4],
[4, 2, 1, 3], and [4, 1, 3, 2]. Furthermore, one can check that these are the only permutations
p
√
4
for which LR
4 (π, ± 2 + 5) comprises a set of 16 equiangular lines in C for this R. All 8
of these permutations lead to maximum-sized sets of lines that are known via the fiducial
vector construction in the new basis given by Appleby et al. [4] (see §2.7).
We now return to the question of whether or not the construction suggested by The-
orem 38 can be used in dimensions d > 4. First we note that not all almost flat fiducial
vector solutions can be interpreted as coming from a set of d MUBs. For example, Appleby’s
solution [1] for d = 7 does not reduce to a set of 7 MUBs. Furthermore, this construction
must also be limited to prime power dimensions, as (by current knowledge) we can find a
(d, d, d, 1)-RDS to produce d MUBs as in Theorem 17 only for such dimensions. Finally,
computer searches for d = 8, 9 suggest that it may not be possible to find a triple (R, π, v)
such that LR
d (π, v) is a maximum-sized set of equiangular lines in these dimensions.
Note that if we consider only a single basis from a set of MUBs, this is already a set of d
equiangular lines in Cd , albeit unextendable to d2 lines (since the value of a2 from (2.6) is 0
rather than
1
d+1 ,
as required by Proposition 9). Furthermore, multiplying a single entry of
each basis element by some complex constant maintains equiangularity, while allowing for
the value of a2 to change. Noting, as after (2.22), that a single such basis corresponds to a
complex Hadamard matrix brings us to the next construction.
3.2
Construction from a Complex Hadamard Matrix
Our second construction uses d copies of a single complex Hadamard matrix to form d2
equiangular lines in Cd . Similar to the construction in §3.1, we multiply a single entry of
each column vector by some complex constant. This construction gives maximum-sized sets
of almost flat equiangular lines for dimensions 2, 3 and 8. Unlike the examples given in
64
CHAPTER 3. NEW CONSTRUCTIONS
§3.1, which were all previously explained via the fiducial vector method, the example given
here for d = 8 was previously unknown and appears to be the simplest known set of 64
equiangular lines in C8 .
Recall from (2.22) that a complex d × d matrix X having all entries of magnitude 1
is a complex Hadamard matrix if X † X = dId , which implies that the columns of X are
√
orthogonal with norm d. Represent the columns of a d × d complex Hadamard matrix
as a block of vectors and write Xj (v) to mean the block of vectors formed by multiplying
entry j of each vector of the block representing X by some complex constant v. Let X(v) =
∪dj=1 Xj (v).
Example 47. Let ω = e2πi/3 .

1 1 ω


X= 1 ω 1
1 ω2 ω2
Then



(v
Take the complex Hadamard matrix


T


 (1 1 1 )

 whose block of vectors is
(1 ω ω 2 )T



 (ω 1 ω 2 )T



1 )T 
1 )T



 (1 v
X1 (v) = ( v ω ω 2 )T , X2 (v) = (1 ωv ω 2 )T





 (vω 1 ω 2 )T 

 (ω v ω 2 )T
1
and
X(v) =


(v





(v





(vω









(1
(1




(ω








(1





(1




(ω
1
ω
1
v
ωv
v
1
ω
1










 (1
.

v )T 


2
T
, X3 (v) = (1 ω ω v)
,






 (ω 1 ω 2 v)T 


1 )T 



2
T

ω ) 




2
T
ω ) 







T
1 ) 


ω 2 )T














T

v ) 



2
T

ω v) 




2
T
ω v)
1
.
ω 2 )T
Notice that there are only the following three types of inner product that can arise
between distinct vectors of X(v):
65
CHAPTER 3. NEW CONSTRUCTIONS
(i) the inner product of two distinct vectors within a block Xj (v),
(ii) the inner product of two vectors of distinct blocks Xj (v), Xk (v) which are derived from
the same column of X,
(iii) the inner product of two vectors of distinct blocks Xj (v), Xk (v) which are derived from
distinct columns of X.
Considering (i) and (ii), we get the following lemma:
Lemma 48. Let X be a d × d complex Hadamard matrix for d ≥ 2. If v = a + ib,
a, b ∈ R, is a constant depending on d such that X(v) is a set of d2 equiangular lines, then
|a2 + b2 − 1| = |2a + d − 2|.
Proof. Every inner product of the form (i) arises from two vectors which are derived from
distinct columns of X. By (2.22), their inner product in X is 0; thus their inner product in
X(v) has magnitude |v|2 − 1 = |a2 + b2 − 1|.
Every inner product of the form (ii) arises from two vectors which are derived from the
same column of X. By (2.22), their inner product in X is d; thus their inner product in
X(v) has magnitude |v + v + d − 2| = |2a + d − 2|.
The fact that these vectors belong to a set of equiangular lines means these inner product
magnitudes must be equal, giving the result.
We now consider three specific types of complex Hadamard matrix and determine when
we can use them to construct maximum-sized sets of equiangular lines.
Theorem 49. Let X be a d×d real Hadamard matrix. Then there exists a complex constant
v(d) such that X(v(d)) is a set of d2 equiangular lines if and only if d = 2, 8.
Proof. Let X = (xjk ) be a real Hadamard matrix and v = a + ib. Form the set of vectors
X(v) = ∪dj=1 Xj (v) from blocks of vectors Xj (v) derived from X.
Let d = 2. Since
X=
x11 x12
x21 x22
!
,
the only inner products of the form (iii) must arise from a vector derived from column 1
with a vector derived from column 2, or vice-versa. Since X is a real Hadamard matrix
we have x11 x12 + x21 x22 = 0 by (2.22), so that x11 x12 = −x21 x22 . Then the corresponding
inner products in X(v) have magnitude |v − v| = |2b|.
CHAPTER 3. NEW CONSTRUCTIONS
66
Using Lemma 48, X(v) is a set of equiangular lines if and only if we can find v = a + ib
that satisfies
|a2 + b2 − 1| = |2a| = |2b|.
√
√ This is solved by (a, b) = (a, ±a) where a ∈ 12 (1 ± 3), − 12 (1 ± 3) , and does not depend
on our choice of X.
Next suppose d > 2. Then every two vectors whose inner product is of the form (iii)
are derived from distinct columns of X and thus their inner product in X is 0 by (2.22).
Since X is a real Hadamard matrix, every inner product of distinct columns has the form
n(1) + n(−1) for some integer n, and n ≥ 2 since d > 2. In particular, for every pair (s, t)
with s 6= t, there exist distinct 1 ≤ j, k, ` ≤ d such that xjs xjt = xks xkt = −x`s x`t . Thus
between vector s of Xj (v) and vector t in Xk (v) we have an inner product of magnitude
|v +v −2| = |2a−2| and between vector s of block Xj (v) and vector t of block X` (v) we have
an inner product of magnitude |v − v| = |2b|. Furthermore since every xjs xjt ∈ {1, −1}, for
every s 6= t and j 6= k we have xjs xjt = ±xks xkt , so that every inner product of the form
(iii) has one of these two magnitudes.
Then using Lemma 48, X(v) is a set of equiangular lines if and only if we can find
v = a + ib that satisfies
|a2 + b2 − 1| = |2a + d − 2| = |2a − 2| = |2b|.
From |2a − 2| = |2b| we get b = ±(a − 1). Thus we must have
|a2 + (a − 1)2 − 1| = |2(a − 1)|
which implies a = ±1. We cannot have a = 1, otherwise b = 0, and then |2(1)+d−2| = |2(0)|
leaving us with d = 0. Thus a = −1 giving b = ±2 and |2(−1) + d − 2| = |2(±2)|, which
since d > 2 is solved exactly when d = 8.
Example 50. d = 2 : Take the real Hadamard matrix
!
1 1
X=
1 −1
67
CHAPTER 3. NEW CONSTRUCTIONS
and v = a + ib. Then our construction gives the set of vectors


T 

(a
+
ib
1
)








T

 (a + ib

−1 ) 
X(v) =



a + ib )T 
( 1







( 1
T
−(a + ib)) 
which are equiangular for (a, b) = (a, ±a) where a ∈
Example 51. d = 8 : Take the

1


 1



 1


 1

X=

 1



 1


 1


1
1
2 (1 ±
√
3), − 21 (1 ±
real Hadamard matrix
1
1
1
1
1
1
1
−1
1
−1
1
−1
1
−1
−1 −1
1
1
1
−1 −1
1
1
−1
1
1
1
1
1
−1 −1
−1 −1
1
1
−1 −1 −1 −1
1
−1 −1 −1 −1
−1 −1
−1 −1
−1
1
−1
1
1
1
1
−1
√ 3) .












.











Then X(a + ib) is a set of 64 equiangular lines for a = −1, b = ±2.
Notice that Theorem 49 can fail for d = 8 if we remove the restriction that X be
real. To see this, take X = (xjk ) to be the Fourier matrix; that is xjk = e2πi(j−1)(k−1)/8 for
j, k ∈ {1, . . . , 8}. It is enough to look at the inner products between X1 (a+ib) and X2 (a+ib),
in particular the vectors derived from column 1 of X in X1 (a + ib) with those derived from
√
columns 3, 5, 7 of X in X2 (a + ib), to find that one obtains magnitudes 2|a − 1 − b|, 2|b|
√
and 2|a − 1 + b|. Equating these requires a = 1, b = 0 and leads to inner products of 0
rather than of magnitude
√1
d+1
as required by Proposition 9. We will see that these same
magnitudes arise again in the proof of Theorem 54.
Theorem 52. Let X be a d × d complex Hadamard matrix all of whose entries are third
roots of unity. Then there exists a complex constant v(d) such that X(v(d)) is a set of d2
equiangular lines if and only if d = 3.
68
CHAPTER 3. NEW CONSTRUCTIONS
Proof. Let ω = e2πi/3 . Let X = (xjk ) be a complex Hadamard matrix with each xjk ∈
{1, ω, ω 2 } and v = a + ib. Form the set of vectors X(v) = ∪dj=1 Xj (v) from blocks of vectors
Xj (v) derived from X.
Take d = 3. Since

x11 x12 x13




X=
 x21 x22 x23  ,
x31 x32 x33
the inner product of every pair of distinct columns of X must have the form 1 + ω + ω 2 = 0
by (2.22). Thus every two vectors having inner product of the form (iii) have inner product
1 + ω + ω 2 = 0 in X. In particular, for every pair (s, t) with s 6= t, we never have
xjs xjt = xks xkt for j 6= k, and for {j, k, `} = {1, 2, 3} we have xjs xjt = ωxks xkt = ω 2 x`s x`t .
Thus between vector s in block Xj (v) and vector t in block Xk (v) we have an inner product
√
of magnitude |v+vω 2 +ω| = |a−1−b 3|, and between vector s in block Xj (v) and vector t in
√
block X` (v) we have an inner product of magnitude |v+vω+ω 2 | = |a−1+b 3|. Furthermore,
since every xjs xjt ∈ {1, ω, ω 2 }, for every s 6= t and j 6= k we have xjs xjt = ω ` xks xkt for
some ` ∈ {1, 2}, so every inner product of the form (iii) has one of these two magnitudes.
Then using Lemma 48, X(v) is a set of equiangular lines if and only if we can find
v = a + ib that satisfies
√
√
|a2 + b2 − 1| = |2a + 1| = |a − 1 + b 3| = |a − 1 − b 3|.
√
√
From |a − 1 + b 3| = |a − 1 − b 3|, we see that a = 1 or b = 0. Now when a = 1 we have
√
|b2 | = |3| = |b 3|,
√
which is solved by b = ± 3. When b = 0 we have
|a2 − 1| = |2a + 1| = |a − 1|,
which is solved by a = −2, 0. Thus X(v) is a set of 9 equiangular lines in C3 when v ∈
√
{0, −2, 1 ± i 3}, regardless of our choice of X.
Next suppose d > 3. Then every pair of distinct columns of X has inner product
of form n(1 + ω + ω 2 ) = 0 by (2.22), for n ≥ 2 since d > 3; thus every two vectors
with inner product of the form (iii) have inner product n(1 + ω + ω 2 ) = 0 in X. In
particular, for every pair (s, t) with s 6= t, there exist distinct 1 ≤ j, k, `, m ≤ d such that
xms xmt = xjs xjt = ωxks xkt = ω 2 x`s x`t . Thus between vector s in block Xj (v) and vector t
69
CHAPTER 3. NEW CONSTRUCTIONS
√
in block Xk (v) we have an inner product of magnitude |v + vω 2 + ω| = |a − 1 − b 3|, between
vector s in block Xj (v) and vector t in block X` (v) we have an inner product of magnitude
√
|v + vω + ω 2 | = |a − 1 + b 3|, and between vector s in block Xj (v) and vector t in block
Xm (v) we have an inner product of magnitude |v + v − 2| = |2a − 2|. Furthermore, since
every xjs xjt ∈ {1, ω, ω 2 }, for every s 6= t and j 6= k we have xjs xjt = ω ` xks xkt for some
` ∈ {0, 1, 2}, so every inner product of the form (iii) has one of these three magnitudes.
Then using Lemma 48, X(v) is a set of equiangular lines if and only if we can find
v = a + ib that satisfies
√
√
|a2 + b2 − 1| = |2a + d − 2| = |2a − 2| = |a − 1 + b 3| = |a − 1 − b 3|.
√
√
Again, from |a − 1 + b 3| = |a − 1 − b 3| we see that a = 1 or b = 0. From |2a − 2| =
√
|a − 1 + b 3| we see that we must have a = 1 and b = 0. However, |2(1) + d − 2| = |2(1) − 2|
is not solved by any d > 3.
Example 53. d = 3 : Take

1
1
ω




X=
 1 ω 1 
1 ω2 ω2
and v = a + ib. Then our construction gives the set of vectors


( a + ib
1
1





( a + ib
ω
ω2





((a + ib)ω
1
ω2








1
a + ib
1

(
X(v) =
(




(







(





(




(
1
ω
1
1
ω

)T 



T

) 




T
) 







T
) 


2
T
(a + ib)ω
ω
)


2

a + ib
ω
)T 






T

1
a + ib ) 



2
T

ω
(a + ib)ω ) 




2
T
1
(a + ib)ω )
√
which are equiangular for a + ib ∈ {0, −2, 1 ± i 3}.
The following theorem generalizes Theorem 49.
70
CHAPTER 3. NEW CONSTRUCTIONS
Theorem 54. Let X be a d × d complex Hadamard matrix all of whose entries are fourth
roots of unity. Then there exists a complex constant v(d) such that X(v(d)) is a set of d2
equiangular lines if and only if d = 2, 8.
Proof. Let X = (xjk ) be a complex Hadamard matrix with each xjk ∈ {1, −1, i, −i} and
v = a + ib. Form the set of vectors X(v) = ∪dj=1 Xj (v) from blocks of vectors Xj (v).
Every pair of distinct columns in X has inner product of the form n(1) + n(−1) + m(i) +
m(−i) = 0 for some integers n, m. Thus every two vectors with inner product of the form
(iii) have inner product n(1) + n(−1) + m(i) + m(−i) = 0 in X.
We consider two cases. We begin with the case where there is some inner product of the
form (iii) with inner product in X having m, n ≥ 1. This means there is some pair (s, t)
with s 6= t, such that there exist distinct 1 ≤ j, k, `, p ≤ d with xjs xjt = −xks xkt = ix`s x`t =
−ixps xpt . Thus between vector s in block Xj (v) and vector t in block Xk (v) we have an
inner product of magnitude |v − v| = |2b|, between vector s of block Xj (v) and vector t of
√
block X` (v) we have an inner product of magnitude |i(v − 1) + (v − 1)| = 2|a − 1 − b| and
between vector s of block Xj (v) and vector t of block Xp (v) we have an inner product of
√
magnitude | − i(v − 1) + (v − 1)| = 2|a − 1 + b|.
Then using Lemma 48, X(v) is a set of equiangular lines only if we can find v = a + ib
that satisfies
From
√
|a2 + b2 − 1| = |2a + d − 2| = |2b| =
2|a − 1 + b| =
√
√
2|a − 1 + b| =
√
2|a − 1 − b|.
2|a − 1 − b|, we see that a = 1 or b = 0. Since b = 0 implies
all inner products are 0, we cannot have this by Proposition 9. Thus a = 1, and from
√
√
2|a − 1 + b| = |2b|, we have 2|b| = |2b|, which is only solved by b = 0, and we have
already seen this is not an acceptable solution. So in this case there is no suitable v.
This leaves the second case where every inner product of the form (iii) has inner product
in X with one of m, n being 0. In this case we claim that we can transform X into a real
Hadamard matrix by a sequence of row and column multiplications by i. We can then apply
Theorem 49 and transform back by reversing the sequence of operations to obtain the result.
It remains to prove the claim. We have that every inner product of distinct columns
of X is either n(1) + n(−1) or m(i) + m(−i). Multiplying a row or column of X by i
does not change the form of such an inner product and preserves the complex Hadamard
property. Therefore, we may transform the first column of X to have all entries in {1, −1}
(by a sequence of row multiplications). Now every column of X whose inner product with
71
CHAPTER 3. NEW CONSTRUCTIONS
the first column has the form n(1) + n(−1) contains entries which are only in {1, −1}.
The remaining columns of X have inner product with the first column of X of the form
m(i)+m(−i) and must therefore contain entries which are only in {i, −i}. We now transform
X to be a real Hadamard matrix by multiplying all such columns by i.
Example 55. d = 2 : Take
X=
1
1
i −i
!
and v = a + ib. Notice we can transform X to a real Hadamard matrix by multiplying the
second row by i. (In fact, we obtain the real Hadamard matrix of Example 50 if we multiply
by −i) . Then our construction gives the set of vectors


i
)T
 (a + ib




 (a + ib
−i
)T
X(v) =


( 1
i(a + ib) )T




( 1
−i(a + ib))T
which are equiangular for (a, b) = (a, ±b) where a ∈
1
2 (1
±















√
3), − 12 (1 ±
√ 3) .
Notice that in this construction every block of d vectors is derived from the same initial
matrix; thus given a particular d × d complex Hadamard matrix X it is quite simple to
determine if there is a constant v such that X(v) is a set of d2 equiangular lines. Furthermore,
it is particularly easy to verify that the vectors given in Examples 50 and 51 form maximumsized sets of equiangular lines, since the initial Hadamard matrix is real. Examples 53 and
55 are not much harder to check.
We further show the dimensions 2, 3 and 8 specified in Theorems 49, 52 and 54 are the
only ones for which this construction produces maximum-sized sets of equiangular lines.
Theorem 56. Let d ≥ 2 and suppose X(v(d)) is a set of d2 equiangular lines for some
complex Hadamard matrix X and complex constant v(d). Then d ∈ {2, 3, 8}.
Proof. Let X = (xjk ) be a complex Hadamard matrix and v = a + ib. Form the set of
vectors X(v) = ∪dj=1 Xj (v) from blocks of vectors Xj (v). We will show that if X(v) is a set
of d2 equiangular lines for d ≥ 4, then d = 8.
72
CHAPTER 3. NEW CONSTRUCTIONS
Every pair of distinct columns of X has inner product 0 by (2.22). Thus every two
vectors in X(v) with inner product of the form (iii) have inner product 0 in X.
We consider two cases. We begin with the case where the inner product of every pair of
distinct columns in X has the form n(ξ)+n(−ξ) for some ξ ∈ C of magnitude 1. Multiplying
a row or column of X by any complex constant of magnitude 1 does not change the form
of such an inner product and preserves the complex Hadamard property. Therefore, we
may transform the first column of X to have all entries in {1, −1} (by a sequence of row
multiplications). Now for each j 6= 1, column j of X has inner product n(ξj ) + n(−ξj ) with
the first column for some ξj ∈ C and therefore contains entries which are only in {ξj , −ξj }.
We now transform X to be a real Hadamard matrix by multiplying each column j by ξj−1 .
We now apply Theorem 49 to find that we must have d = 8 and then transform back by
reversing the sequence of multiplications.
This leaves the second case where some pair of distinct columns of X has inner product
P
of the form tj=1 nj ξj for distinct ξj ∈ C of magnitude 1, where t ≥ 3 and each nj ≥ 1.
Suppose that ξ1 , ξ2 , ξ3 arise from xjs xjt , xks xkt , x`s x`t , respectively, for distinct j, k, ` and
/ {j, k, `}. Thus
s 6= t. Furthermore, since d ≥ 4, there is also some xms xmt = ζ with m ∈
between vector s in block Xm (v) and vector t in block Xj (v) we have an inner product of
(a + ib − 1)ζ + (a − ib − 1)ξ1 , between vector s in block Xm (v) and vector t in block Xk (v)
we have an inner product of (a + ib − 1)ζ + (a − ib − 1)ξ2 and between vector s in block
Xm (v) and vector t in block X` (v) we have an inner product of (a + ib − 1)ζ + (a − ib − 1)ξ3 .
For X(v) to be a set of equiangular lines, we require
|(a + ib − 1)ζ + (a − ib − 1)ξ1 | = |(a + ib − 1)ζ + (a − ib − 1)ξ2 |
= |(a + ib − 1)ζ + (a − ib − 1)ξ3 |
(3.11)
Notice that from Lemma 48, we cannot have (a, b) = (1, 0) as this implies d = 0. Thus
p
(a + ib − 1)ζ 6= 0 and (a − 1)2 + b2 6= 0. Now all the ξj are distinct with magnitude 1, so
p
(a−ib−1)ξ1 , (a−ib−1)ξ2 , (a−ib−1)ξ3 are all distinct with magnitude (a − 1)2 + b2 > 0,
but then only two of them when summed with (a + ib − 1)ζ can have equal magnitude, as
illustrated below.
73
CHAPTER 3. NEW CONSTRUCTIONS
=
radius
p
(a − 1)2 + b2
(a + ib − 1)ζ
<
3.3
Construction from Building Blocks in a Smaller Dimension
Our final construction exhibits a connection between sets of equiangular lines in different
dimensions. In particular, we construct another new set of 64 equiangular lines in C8 from
building blocks inspired by equiangular lines in C4 .
Recall from Example 41 that for a particular RDS R and certain permutations π, the
p
√
4
set LR
4 (π, ± 2 + 5) comprises 16 equiangular lines in C ; furthermore, the proof of The-
orem 46 illustrates that with the correct permutations the magnitudes of inner products of
elements of LR
4 (π, v) are highly constrained. We now show how we can combine 4 sets of
8
vectors of the form LR
4 (π, v) to form 64 vectors in C which also have highly constrained
inner product magnitudes.
As described in §3.1, the vectors in LR
d (π, v) are derived from vectors in a set of d MUBs
R 0 0
constructed from an RDS. For an RDS R, write [LR
d (π, v) Ld (π , v )] for the set of vectors
R 0 0
which are the concatenation of corresponding vectors in LR
d (π, v) and Ld (π , v ).
74
CHAPTER 3. NEW CONSTRUCTIONS
Example 57. d = 4 :Take the (4, 4, 4, 1)-RDS, R = {1, x, y, x3 y 3 } in G = hxi×hyi = Z4 ×Z4
relative to hx2 i × hy 2 i ∼
= Z2 × Z2 , as in Example 18. Take permutations π = [1, 3, 4, 2], π 0 =
R 0 0
[1, 4, 3, 2] and constants v, v 0 . Construct LR
4 (π, v), L4 (π , v ) as in (3.1) (see Example 41).
R 0 0
Then form [LR
4 (π, v) L4 (π , v )] as follows:
R 0 0
[LR
4 (π, v) L4 (π , v )] =


(v





(v





(v






(v








(1





(1





(1





 (1
1
1
1
v0
1
1
1
−1
−1
v0
1
−1
1
v0
−1
−1
−1
−1
v0
1
iv
1
−iv
−i
−1
−1
−1


(1
i






(1
i





(1 −i






(1 −i







(1 iv






(1 iv





(1 −iv




(1 −iv
1
1
−1
−1
1
1
i
i
1
1
iv
i
1
−iv
−i
1
−1
−i
−1
−i
i
v0
−v 0
−iv
1
iv
1
i
1
iv
1
−1
−iv
1
−i
i
−1
1
−1
i

1 )T 



T

−1 ) 




T
−1 ) 




T

1 ) 






0
T

−iv ) 



0
T

iv ) 




0
T
iv ) 




0
T

−iv ) 
−i
)T
i )T
v0
i )T
−i
−v 0
−i )T
1
iv 0
i
1
1
iv 0
i
1
1
−i
−1
1
−iv 0
−i
−1 )T
−i
−iv 0
1 )T
i
1 )T
−i
−1 )T









































.
The next lemma shows how to combine two sets of LR
d (π, v) so that their inner product
magnitudes are highly constrained.
R
2d have all inner products
Lemma 58. The d2 vectors of [LR
d (π, a+ib) Ld (π, 2−a−ib)] in C
√
of magnitude 2(b2 + (a − 1)2 ) or 2 d between distinct vectors.
Notice that unlike in Example 57, Lemma 58 requires that the permutation π be the
same in both sets of the smaller dimensional building blocks.
Proof. Recall that the vectors of LR
d (π, v) are formed from vectors of d MUBs formed from
a (d, d, d, 1)-RDS R, as in Theorem 17, and their entries are roots of unity.
75
CHAPTER 3. NEW CONSTRUCTIONS
We begin by considering the inner product of distinct vectors constructed from vectors
in the same block BjR (π, v) of LR
d (π, v). Since the original vectors are orthogonal, we can see
that this inner product is ξ(|v|2 − 1), for some root of unity ξ. When v = a + ib, the inner
product becomes ξ(a2 + b2 − 1) and when v = 2 − a − ib it becomes ξ((2 − a)2 + b2 − 1). Thus
the corresponding concatenated vectors have inner product ξ(a2 + b2 − 1 + (2 − a)2 + b2 − 1),
which has magnitude 2(b2 + (a − 1)2 ).
Consider vectors in distinct blocks BjR (π, v), BkR (π, v) of LR
d (π, v). Let these vectors be
given by
x = (x1 x2 . . .
...
vxπ(j) . . .
xd )T
(3.12)
. . . yd )T .
P
Their inner product before multiplying certain elements by v is d`=1 x` y` , which by (2.24)
√
has magnitude d. Now the inner product of x, y in LR
d (π, v) is
y = (y1
y2 . . . vyπ(k)
x1 y1 +· · ·+vxπ(j) yπ(j) +vxπ(k) yπ(k) +· · ·+xd xd =
d
X
...
x` y` +(v−1)xπ(j) yπ(j) +(v−1)xπ(k) yπ(k) .
`=1
This means that our corresponding concatenated vectors have inner product
d
X
`=1
+
x` y` + (a + ib − 1)xπ(j) yπ(j) + (a − ib − 1)xπ(k) yπ(k)
d
X
`=1
x` y` + (2 − a − ib − 1)xπ(j) yπ(j) + (2 − a + ib − 1)xπ(k) yπ(k) = 2
d
X
x ` y`
`=1
P
√
√
Since d`=1 x` y` = d, our concatenated vectors have inner product of magnitude 2 d,
as claimed.
√
Notice that we can find a, b such that 2(b2 + (a − 1)2 ) = 2 d for any dimension d.
Thus Lemma 58 gives d2 equiangular lines in C2d whenever d = pr for some prime p, as we
can construct blocks LR
d (π, v) in these dimensions. So the list of dimensions for which one
can construct Θ(d2 ) equiangular lines, which was previously known to include d = pr (see
Theorem 19), d = 3 · 22t−1 − 1 [17], and d = pr + 1, for p prime [36], can now be extended
to include d = 2pr for p prime.
However, we wish to construct the maximum number 4d2 equiangular lines in C2d .
Recall we know from Proposition 9 that distinct vectors x, y ∈ C2d in a maximum-sized set
of equiangular lines must satisfy
1
|hx, yi|
=√
.
||x|| · ||y||
2d + 1
76
CHAPTER 3. NEW CONSTRUCTIONS
R
The norm of each vector in [LR
d (π, a + ib) Ld (π, 2 − a − ib)] is
p
2((a − 1)2 + b2 + d), so that
if we hope to extend the set of lines given by Lemma 58, a, b must satisfy
√
2 d
1
2(b2 + (a − 1)2 )
=
=√
.
2((a − 1)2 + b2 + d)
2((a − 1)2 + b2 + d)
2d + 1
We can solve these equations to find that they can be satisfied only when d = 4. We now
show how the set of lines given by Lemma 58 for d = 4 can be extended to a maximum-sized
set of equiangular lines in C8 .
Fix the (4, 4, 4, 1)-RDS R and permutation π ∈ S4 . In order to find constants v such
that we can extend the d2 lines given by Lemma 58 for d = 4 to 4d2 equiangular lines, we
impose additional structure on the LR
4 blocks. For example, one might first assume that 64
lines should be created as
R
[LR
4 (π, a1 + ib1 ) L4 (π, 2 − a1 − ib1 )]
R
[LR
4 (π, a2 + ib2 ) L4 (π, 2 − a2 − ib2 )]
R
[LR
4 (π, a3 + ib3 ) L4 (π, 2 − a3 − ib3 )]
R
[LR
4 (π, a4 + ib4 ) L4 (π, 2 − a4 − ib4 )].
However one quickly finds that there is no set of real aj , bj for which this is a set of 64
equiangular lines in C8 . There are aj , bj which yield a set of 32 equiangular lines (with each
line duplicated). Inspection of the aj , bj reveals that these 32 lines arise from the structure
[LR
4 (π,
a1 + ib1 ) LR
4 (π, 2 − a1 − ib1 )]
R
[LR
4 (π, 2 − a1 − ib1 ) L4 (π,
[LR
4 (π,
[LR
4 (π,
a2 + ib2 )
2 − a2 − ib2 )
LR
4 (π,
LR
4 (π,
a1 + ib1 )]
2 − a2 − ib2 )]
a2 + ib2 )].
Working from this structure, we seek to make small changes to one or more of the LR
4
blocks that maintain the highly constrained inner products without duplicating lines. If we
have chosen an appropriate permutation π, then we quickly find that possibly the simplest
change, namely negating two identical LR
4 blocks, admits a1 , a2 , b1 , b2 and π which give a
set of 64 equiangular lines in C8 .
Example 59. Let R be the RDS of Example 18 and π = [1, 3, 4, 2]. The following is a set
77
CHAPTER 3. NEW CONSTRUCTIONS
of 64 equiangular lines in C8 :
[ LR
4 (π,
[ LR
4 (π,
[ LR
4 (π,
[−LR
4 (π,
2 + i)
LR
4 (π,
−i )
LR
4 (π,
−LR
4 (π,
LR
4 (π,
−1 + 2i )
1)
−i ) ]
2 + i)]
1)]
−1 + 2i ) ]
The lines are given explicitly as the transposed rows of

2+i
1
1
1
−i
1
1 − 2i
−i
−1
1
1
1
1





 2+i
1
−1
−1
−i 1 −1 −1 




 2+i
−1
1
−1
−i −1 1 −1 






 2+i

−1
−1
1
−i
−1
−1
1






1
1
−1
+
2
i
−i
1
1
1
−i






 1
1
1 − 2i
i
1
1 −1 i 






 1
−1
−1 + 2 i
i
1 −1 1
i 




 1

−1
1
−
2
i
−i
1
−1
−1
−i






 1
i
1
1 − 2i
1
i
1 −1 






i
−1
−1 + 2 i 1
i −1 1 
 1




 1

−i
1
−1
+
2
i
1
−i
1
1






 1
−i
−1
1 − 2i
1 −i −1 −1 




 1
−1 + 2 i
i
−1
1
1
i −1 






 1

−1
+
2
i
−i
1
1
1
−i
1






1
1
−
2
i
i
1
1
−1
i
1




−1 −i −1
78
CHAPTER 3. NEW CONSTRUCTIONS




















































−i
1
−i
1
−i −1
1
2+i
1
1
1
−1 −1 2 + i
1
−1
−1
−1
1
−1
1
−i −1 −1
1
−1 2 + i
1
2+i
−1
−1
1
1
1
1
−i
1
1
−1 + 2 i
−i
1
1
−1
i
1
1
1 − 2i
i
1
−1
1
i
1
−1
−1 + 2 i
i
1
−1 −1 −i
1
−1
1 − 2i
−i
1
i
1
−1
1
i
1
1 − 2i
1
i
−1
1
1
i
−1
−1 + 2 i
1
−i
1
1
1
−i
1
−1 + 2 i
1
−i −1 −1
1
−i
−1
1 − 2i
1
1
i
−1
1
−1 + 2 i
i
−1
1
1
−i
1
1
−1 + 2 i
−i
1
1
−1
i
1
1
1 − 2i
i
1
1
−1 −i −1
1
1 − 2i
−i
−1




















































79
CHAPTER 3. NEW CONSTRUCTIONS

−1 + 2 i
1
1
1


 −1 + 2 i
1
−1
−1


 −1 + 2 i
−1
1
−1



 −1 + 2 i
−1
−1
1



1
1
−2 − i
−i




1
1
2+i
i




1
−1
−2 − i
i



1
−1
2+i
−i




1
i
1
2+i



1
i
−1
−2 − i




1
−i
1
−2 − i




1
−i
−1
2+i



1
−2 − i
i
−1




1
−2 − i
−i
1



1
2+i
i
1


1
2+i
−i
−1
−1 −1 −1 −1
−1 −1
1
1
−1
1
−1
1
−1
1
1
−1
−1 −1 −i
−1 −1
i
i
−i
−1
1
−i
−i
−1
1
i
i
−1 −i −1
−1 −i
−1
i
−1
i
1
i
−i
−1 −i
1
i
−1 −i
−i
1
−1 −i
i
−1
−1
i
−1
i
−i −1
i
1




















































80
CHAPTER 3. NEW CONSTRUCTIONS




















































−1 −1 −1 −1 −1 + 2 i
1
1
1
−1 −1
1
1
−1 + 2 i
1
−1
−1
1
−1 + 2 i
−1
1
−1
−1 −1 + 2 i
−1
−1
1
−1
1
−1
−1
1
1
−1 −1 −i
−1 −1
i
1
1
−2 − i
−i
i
−i
1
1
2+i
i
−1
1
−i
−i
1
−1
−2 − i
i
−1
1
i
i
1
−1
2+i
−i
i
1
i
1
2+i
−i
1
i
−1
−2 − i
−1 −i
1
−i
1
−2 − i
−1 −i −1
−1 −i
−1
i
−1
i
1
1
i
1
−i
−1
2+i
−1 −i
−i
1
1
−2 − i
i
−1
−1 −i
i
−1
1
−2 − i
−i
1
−i −1
1
2+i
i
1
1
2+i
−i
−1
−1
i
−1
i
i
1
Notice that this is also an example of almost flat equiangular lines.


























.

























This solution is one of the following four solutions based on this RDS R and permutation
π that were found using the computer algebra system Maple:
[ LR
4 (π,
[ LR
4 (π,
[ LR
4 (π,
[−LR
4 (π,
2 + i)
LR
4 (π,
−i )
LR
4 (π,
−LR
4 (π,
LR
4 (π,
−1 + 2i )
1)
−i ) ]
2 + i)]
1)]
−1 + 2i ) ]
81
CHAPTER 3. NEW CONSTRUCTIONS
[ LR
4 (π, 2 + i )
[ LR
4 (π,
[ LR
4 (π,
[−LR
4 (π,
LR
4 (π,
1 − 2i )
LR
4 (π,
−LR
4 (π,
LR
4 (π,
[ LR
4 (π, 2 − i )
LR
4 (π,
[ LR
4 (π,
−i )
−1 )
i)
[ LR
4 (π,
[ LR
4 (π,
[ LR
4 (π,
[−LR
4 (π,
2 + i)]
−1 ) ]
1 − 2i ) ]
i)]
LR
4 (π, 2 − i ) ]
R
[ LR
4 (π, 1 + 2i ) −L4 (π,
[−LR
4 (π,
−i ) ]
−1 ) ]
−1 )
LR
4 (π,
2 − i)
LR
4 (π,
i)]
LR
4 (π,
−LR
4 (π,
LR
4 (π,
2 − i)]
i)
−1 − 2i )
1)
1 + 2i ) ]
1)]
−1 − 2i ) ]
R 0
A natural question that arises is if one must construct [LR
4 (π, a + ib) L4 (π , 2 − a − ib)]
with π 0 = π. Looking at the inner products that arise from π 0 6= π, it seems that the answer
should be yes. Otherwise, the magnitudes of these inner products are no longer tightly
constrained as in Lemma 58. As previous noted, the particular choice of π also turns out
to be important. If we fix R to be the same RDS as explored in §3.1, given the first 32
R
R
R
vectors, [LR
4 (π, a1 + ib1 ) L4 (π, 2 − a1 − ib1 )] and [L4 (π, 2 − a1 − ib1 ) L4 (π, a1 + ib1 )], the
choice of π does not affect whether or not we can find a1 , b1 such that the result is a set of
32 equiangular lines. However, introducing the second set of 32 vectors, we find that only 8
of the 24 permutations in S4 admit solutions a1 , a2 , b1 , b2 resulting in a set of 64 equiangular
lines. Furthermore, these 8 permutations are the same as the permutations which resulted
p
√
4
in LR
4 (π, ± 2 + 5) comprising a set of 16 equiangular lines in C in §3.1.
The most pressing question that results from considering this construction is whether
there is a way to adapt Lemma 58 to deal with dimensions d 6= 4. Can we use the ideas of
constraining the magnitudes of inner products by constructing lines from building blocks in
smaller dimensions to recursively construct maximum-sized sets of equiangular lines?
Chapter 4
Conclusions and Questions
The fundamental conjecture addressed in this thesis remains open.
Conjecture 2. For each integer d ≥ 2, there is a set of d2 equiangular lines in Cd .
We have synthesized the results obtained by many authors working in several different
paradigms in their own attempts to address Conjecture 2. The landmark results in the
study of equiangular lines can all be easily viewed using the single framework described in
Chapter 2 (See Figures 1.1 and 1.2).
We have also seen the stronger Conjecture 30.
Conjecture 30. For each d ≥ 2, there exists fiducial vectors, {x}, in the eigenspace belonging to the eigenvalue 1 of the d × d matrix Zu , such that
L = {V j U k x : j, k ∈ Zd }
is a set of d2 equiangular lines in Cd .
While assuming that Conjecture 30 holds has allowed many maximum-sized sets of
equiangular lines to be computed, it has also resulted in increasingly complicated sets of
lines. As dimension increases the computational resources required to find such solutions
are becoming unrealistic. The work described in Chapter 2 raises the following questions:
Q1) (§2.6) Does the existence of a numerical set of d2 equiangular lines in Cd imply the
existence of an exact set? And if so, how can we use the knowledge of numerical
solutions to construct exact solutions?
82
CHAPTER 4. CONCLUSIONS AND QUESTIONS
83
Q2) (§2.7) Are there further simplifications to the fiducial vector construction method, such
as yet another basis, which would allow for fiducial vectors to be computed within the
limits of current computational resources?
Setting the fiducial vector method of construction aside, in Chapter 3 we described three
new methods of construction for simple maximum-sized sets of equiangular lines. The first
construction reinterprets known examples for d = 2, 3, 4 via a previously unrecognized connection to MUBs. The second construction constructs maximum-sized sets of equiangular
lines for d = 2, 3, 8 from a complex Hadamard matrix. This construction gives some of the
simplest known sets of d2 equiangular lines in Cd , in particular the set for d = 8 which is a
new example. The third construction builds another new set of 64 equiangular lines in C8
by starting with building blocks that are related to maximum-sized sets of equiangular lines
in C4 . These constructions raise the following questions:
Q3) (§3.1) Are there dimensions d > 4 for which we can construct a set of d2 almost flat
equiangular lines starting from a set of MUBs?
Q4) (§3.1) Are there other known maximum-sized sets of equiangular lines that can be
transformed such that they can be viewed as coming from a set of MUBs?
Q5) (§3.2) Is there an interesting geometric interpretation of the 64 equiangular lines in C8
resulting from the Hadamard matrix construction of §3.2, analogous to the interpreta-
tion of the 64 equiangular lines in C8 given by Hoggar as arising from the diameters of
a polytope [32]?
Q6) (§3.3, Lemma 58) Is there a different way to constrain the magnitudes of inner products between smaller-dimensional building blocks that would allow the construction of
maximum-sized sets of equiangular lines in higher dimensions?
Q7) (§3.3) Is there a different form of building block that one may use to construct maximumsized sets of equiangular lines in higher dimensions?
Q8) (§3.3) Is there a recursive construction of equiangular lines that results in maximumsized sets for an infinite family of dimensions?
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Appendix A
A.1
d = 8 Fiducial Vectors [3, 39]
Scott and Grassl [39] give the following fiducial vector for d = 8 using the standard representation of the Weyl-Heisenberg group.
w2:=sqrt(2);
w3:=sqrt(3);
w5:=sqrt(5);
w6:=w2*w3;
w10:=w2*w5;
w15:=w3*w5;
w30:=w2*w3*w5;
s1:=sqrt(2+w2);
s2:=sqrt(w5-1);
s3:=sqrt(6+w6);
I:=sqrt(-1);
c01:=125008*w5;
c02:=((-1046*w30+7295*w6-691*w10+12550*w2+226*w15+9220*w3-12474*w5-3470)*s1*s2*s3+(74*w30
-6980*w6+12717*w10-24405*w2+10617*w15-28065*w3+14147*w5-30155)*s1*s2+(563*w30-1045*w6-1966*w10
+19858*w2-2983*w15-9011*w3+3131*w5-14565)*s1*s3+(11740*w30-13710*w6-20180*w10+9990*w2-13566*w15
+32610*w3+15732*w5-18120)*s1+(2855*w30-7731*w6-422*w10-18216*w2+10462*w15-2578*w3+1130*w5
-34434)*s2*s3+(-12461*w30+13415*w6-5640*w10+61260*w2-15687*w15+23865*w3-19035*w5+58005)*s2
+(11918*w30-17830*w6+12310*w10-52726*w2+13028*w15-13148*w3+23366*w5-51590)*s3-23811*w30
+52875*w6-57805*w10+130725*w2-47232*w15+71400*w3-55134*w5+151590)*I+(-2136*w30+4365*w6
-7043*w10-4668*w2+1806*w15+2186*w3-256*w5+10990)*s1*s2*s3+(3841*w30+1955*w6+9596*w10-24590*w2
+1319*w15-40355*w3+8647*w5+16925)*s1*s2+(-2346*w30+13014*w6-9557*w10+9135*w2-9871*w15+15525*w3
-13699*w5+34061)*s1*s3+(21494*w30-36960*w6+26942*w10-67440*w2+19504*w15-52620*w3+27674*w5
-47310)*s1+(482*w30-7780*w6+5547*w10-9427*w2+4782*w15-7326*w3+9030*w5-1090)*s2*s3+(-6444*w30
87
APPENDIX A.
88
+21120*w6-10845*w10+21795*w2-10029*w15+8175*w3-25859*w5+86705)*s2+(-730*w30+2162*w6+190*w10
-13406*w2+2194*w15-7578*w3+3496*w5-7232)*s3-18253*w30+28425*w6+5607*w10+5445*w2-3382*w15
+150*w3-7896*w5+35280;
c03:=((9746*w30-1834*w6+7738*w10-28206*w2+8402*w15-17158*w3+1024*w5-41556)*s2*s3+(-20773*w30
+55285*w6-24561*w10+41265*w2-23306*w15+35210*w3-24972*w5+124620)*s2+(6960*w30-10536*w6
+11888*w10-21660*w2+10268*w15-100*w3+14880*w5-86024)*s3-23050*w30+18210*w6-30390*w10+110850*w2
-3420*w15+104280*w3-54336*w5+56340)*I+(1826*w30+13554*w6-11178*w10+3322*w2+904*w15+10436*w3
-3018*w5+13766)*s2*s3+(4503*w30-24915*w6+10601*w10+7555*w2-804*w15-40140*w3+26538*w5-18450)*s2
+(-248*w30-16436*w6-2076*w10+28252*w2-12256*w15-14904*w3+11324*w5+38556)*s3+17974*w30+4470*w6
-19662*w10-26850*w2+34068*w15+53400*w3-12720*w5-97380;
c04:=((-2136*w30+4365*w6-7043*w10-4668*w2+1806*w15+2186*w3-256*w5+10990)*s1*s2*s3+(3841*w30
+1955*w6+9596*w10-24590*w2+1319*w15-40355*w3+8647*w5+16925)*s1*s2+(2346*w30-13014*w6+9557*w10
-9135*w2+9871*w15-15525*w3+13699*w5-34061)*s1*s3+(-21494*w30+36960*w6-26942*w10+67440*w2
-19504*w15+52620*w3-27674*w5+47310)*s1+(-482*w30+7780*w6-5547*w10+9427*w2-4782*w15+7326*w3
-9030*w5+1090)*s2*s3+(6444*w30-21120*w6+10845*w10-21795*w2+10029*w15-8175*w3+25859*w5-86705)*s2
+(-730*w30+2162*w6+190*w10-13406*w2+2194*w15-7578*w3+3496*w5-7232)*s3-18253*w30+28425*w6
+5607*w10+5445*w2-3382*w15+150*w3-7896*w5+35280)*I+(1046*w30-7295*w6+691*w10-12550*w2
-226*w15-9220*w3+12474*w5+3470)*s1*s2*s3+(-74*w30+6980*w6-12717*w10+24405*w2-10617*w15+28065*w3
-14147*w5+30155)*s1*s2+(563*w30-1045*w6-1966*w10+19858*w2-2983*w15-9011*w3+3131*w5-14565)*s1*s3
+(11740*w30-13710*w6-20180*w10+9990*w2-13566*w15+32610*w3+15732*w5-18120)*s1+(2855*w30-7731*w6
-422*w10-18216*w2+10462*w15-2578*w3+1130*w5-34434)*s2*s3+(-12461*w30+13415*w6-5640*w10+61260*w2
-15687*w15+23865*w3-19035*w5+58005)*s2+(-11918*w30+17830*w6-12310*w10+52726*w2-13028*w15
+13148*w3-23366*w5+51590)*s3+23811*w30-52875*w6+57805*w10-130725*w2+47232*w15-71400*w3+55134*w5
-151590;
c05:=((-21172*w30+44012*w6-35588*w10+32212*w2-26080*w15+109776*w3-71128*w5+80168)*s3+(51752*w30
-151440*w6+133800*w10-178920*w2+79104*w15-204600*w3+154080*w5-306360))*I+(-16524*w30+20380*w6
-29948*w10+74324*w2-24216*w15+62472*w3-20240*w5+92480)*s3+55128*w30-106680*w6+65280*w10
-152040*w2+36408*w15-116640*w3+100120*w5-432000;
c06:=((-1046*w30+7295*w6-691*w10+12550*w2+226*w15+9220*w3-12474*w5-3470)*s1*s2*s3+(74*w30
-6980*w6+12717*w10-24405*w2+10617*w15-28065*w3+14147*w5-30155)*s1*s2+(563*w30-1045*w6-1966*w10
+19858*w2-2983*w15-9011*w3+3131*w5-14565)*s1*s3+(11740*w30-13710*w6-20180*w10+9990*w2-13566*w15
+32610*w3+15732*w5-18120)*s1+(-2855*w30+7731*w6+422*w10+18216*w2-10462*w15+2578*w3-1130*w5
+34434)*s2*s3+(12461*w30-13415*w6+5640*w10-61260*w2+15687*w15-23865*w3+19035*w5-58005)*s2
+(-11918*w30+17830*w6-12310*w10+52726*w2-13028*w15+13148*w3-23366*w5+51590)*s3+(23811*w30
-52875*w6+57805*w10-130725*w2+47232*w15-71400*w3+55134*w5-151590))*I+(-2136*w30+4365*w6
-7043*w10-4668*w2+1806*w15+2186*w3-256*w5+10990)*s1*s2*s3+(3841*w30+1955*w6+9596*w10-24590*w2
+1319*w15-40355*w3+8647*w5+16925)*s1*s2+(-2346*w30+13014*w6-9557*w10+9135*w2-9871*w15+15525*w3
-13699*w5+34061)*s1*s3+(21494*w30-36960*w6+26942*w10-67440*w2+19504*w15-52620*w3+27674*w5
-47310)*s1+(-482*w30+7780*w6-5547*w10+9427*w2-4782*w15+7326*w3-9030*w5+1090)*s2*s3+(6444*w30
-21120*w6+10845*w10-21795*w2+10029*w15-8175*w3+25859*w5-86705)*s2+(730*w30-2162*w6-190*w10
+13406*w2-2194*w15+7578*w3-3496*w5+7232)*s3+18253*w30-28425*w6-5607*w10-5445*w2+3382*w15
-150*w3+7896*w5-35280;
c07:=((9746*w30-1834*w6+7738*w10-28206*w2+8402*w15-17158*w3+1024*w5-41556)*s2*s3+(-20773*w30
+55285*w6-24561*w10+41265*w2-23306*w15+35210*w3-24972*w5+124620)*s2+(-6960*w30+10536*w6
89
APPENDIX A.
-11888*w10+21660*w2-10268*w15+100*w3-14880*w5+86024)*s3+(23050*w30-18210*w6+30390*w10-110850*w2
+3420*w15-104280*w3+54336*w5-56340))*I+(1826*w30+13554*w6-11178*w10+3322*w2+904*w15+10436*w3
-3018*w5+13766)*s2*s3+(4503*w30-24915*w6+10601*w10+7555*w2-804*w15-40140*w3+26538*w5-18450)*s2
+(248*w30+16436*w6+2076*w10-28252*w2+12256*w15+14904*w3-11324*w5-38556)*s3-17974*w30-4470*w6
+19662*w10+26850*w2-34068*w15-53400*w3+12720*w5+97380;
c08:=((-2136*w30+4365*w6-7043*w10-4668*w2+1806*w15+2186*w3-256*w5+10990)*s1*s2*s3+(3841*w30
+1955*w6+9596*w10-24590*w2+1319*w15-40355*w3+8647*w5+16925)*s1*s2+(2346*w30-13014*w6+9557*w10
-9135*w2+9871*w15-15525*w3+13699*w5-34061)*s1*s3+(-21494*w30+36960*w6-26942*w10+67440*w2
-19504*w15+52620*w3-27674*w5+47310)*s1+(482*w30-7780*w6+5547*w10-9427*w2+4782*w15-7326*w3
+9030*w5-1090)*s2*s3+(-6444*w30+21120*w6-10845*w10+21795*w2-10029*w15+8175*w3-25859*w5
+86705)*s2+(730*w30-2162*w6-190*w10+13406*w2-2194*w15+7578*w3-3496*w5+7232)*s3+(18253*w30
-28425*w6-5607*w10-5445*w2+3382*w15-150*w3+7896*w5-35280))*I+(1046*w30-7295*w6+691*w10
-12550*w2-226*w15-9220*w3+12474*w5+3470)*s1*s2*s3+(-74*w30+6980*w6-12717*w10+24405*w2-10617*w15
+28065*w3-14147*w5+30155)*s1*s2+(563*w30-1045*w6-1966*w10+19858*w2-2983*w15-9011*w3+3131*w5
-14565)*s1*s3+(11740*w30-13710*w6-20180*w10+9990*w2-13566*w15+32610*w3+15732*w5-18120)*s1
+(-2855*w30+7731*w6+422*w10+18216*w2-10462*w15+2578*w3-1130*w5+34434)*s2*s3+(12461*w30-13415*w6
+5640*w10-61260*w2+15687*w15-23865*w3+19035*w5-58005)*s2+(11918*w30-17830*w6+12310*w10-52726*w2
+13028*w15-13148*w3+23366*w5-51590)*s3-23811*w30+52875*w6-57805*w10+130725*w2-47232*w15
+71400*w3-55134*w5+151590;
phi:=[ c01, c02, c03, c04, c05, c06, c07, c08 ]
Appleby et al. [3] give the following four simpler fiducial vectors for d = 8 using their
k-nomial representation of the Clifford group. Notice that each vector is still decidedly more
complicated than the fiducial vector given in §2.4. Let η = eπi/24 and r = 0, . . . , 3, then the
following vectors |ψi are fiducial:

0


0


 η 33
s
√ 
33
1 3− 5
 η
|ψi =
 √ 15
 2η
2
6



0

 −1

1
A.2


0




0





 η 33
s


√
 ir 1 + 5  η 9


+

 2

6

 √0



 2η 39



 η 12


η 12









.







d = 9 Fiducial Vector [4, 39]
Scott and Grassl [39] give the following fiducial vector for d = 9 using the standard representation of the Weyl-Heisenberg group.
APPENDIX A.
90
w3:=sqrt(3);
w5:=sqrt(5);
w15:=w3*w5;
I:=sqrt(-1);
s1:=1/2*((4+4*I*w3)^(2/3)+4)/(4+4*I*w3)^(1/3);
s2:=((-6+6*I*w5)^(2/3)+6)/(-6+6*I*w5)^(1/3);
s3:=sqrt(1/2*w15+w3);
c01:=240;
c02:=((30*w15*s1-30*w3*s1-30*w5*s1+150*s1)*s3+(30*w5*s1+90*s1))*I+(30*w15*s1-30*w3*s1+30*w5*s1
-150*s1)*s3-30*w15*s1+30*w3*s1;
c03:=((3*w15*s1*s2^2+3*w15*s1*s2-36*w15*s1-3*w15*s2^2-3*w15*s2+36*w15-5*w3*s1*s2^2-5*w3*s1*s2+60*w3*s1
+5*w3*s2^2+5*w3*s2-60*w3+10*w5*s1^2*s2-5*w5*s1*s2-5*w5*s2-30*s1^2*s2+15*s1*s2+15*s2)*s3
-5*w15*s1^2*s2+3/2*w15*s1*s2^2+4*w15*s1*s2-18*w15*s1-3/2*w15*s2^2+w15*s2+18*w15-5*w3*s1^2*s2
-5/2*w3*s1*s2^2+30*w3*s1+5/2*w3*s2^2+5*w3*s2-30*w3+5*w5*s1^2*s2+3/2*w5*s1*s2^2-w5*s1*s2-18*w5*s1
-3/2*w5*s2^2-4*w5*s2+18*w5-15*s1^2*s2+15/2*s1*s2^2+15*s1*s2-90*s1-15/2*s2^2+90)*I+(-5*w15*s1*s2
+5*w15*s2+15*w3*s1*s2-15*w3*s2+6*w5*s1^2*s2^2+6*w5*s1^2*s2-72*w5*s1^2-3*w5*s1*s2^2-3*w5*s1*s2
+36*w5*s1-3*w5*s2^2-3*w5*s2+36*w5-10*s1^2*s2^2-10*s1^2*s2+120*s1^2+5*s1*s2^2+5*s1*s2-60*s1
+5*s2^2+5*s2-60)*s3+w15*s1^2*s2^2+w15*s1^2*s2-12*w15*s1^2-1/2*w15*s1*s2^2-3*w15*s1*s2+6*w15*s1
-1/2*w15*s2^2+2*w15*s2+6*w15+5*w3*s1^2*s2^2+5*w3*s1^2*s2-60*w3*s1^2-5/2*w3*s1*s2^2+5*w3*s1*s2
+30*w3*s1-5/2*w3*s2^2-10*w3*s2+30*w3+3*w5*s1^2*s2^2+3*w5*s1^2*s2-36*w5*s1^2-3/2*w5*s1*s2^2
+6*w5*s1*s2+18*w5*s1-3/2*w5*s2^2-9*w5*s2+18*w5-5*s1^2*s2^2-5*s1^2*s2+60*s1^2+5/2*s1*s2^2
+10*s1*s2-30*s1+5/2*s2^2-5*s2-30;
c04:=(120*w3*s1^2-120*w3*s1-120*w3)*I+120*s1^2-120*s1-120;
c05:=((60*w3*s1^2-60*w3*s1-60*w3-60*w5*s1^2+60*w5*s1+60*w5+120*s1^2-120*s1-120)*s3+(60*w5*s1^2
-60*w5*s1-60*w5))*I+(60*w3*s1^2-60*w3*s1-60*w3+60*w5*s1^2-60*w5*s1-60*w5-120*s1^2+120*s1
+120)*s3+60*w3*s1^2-60*w3*s1-60*w3;
c06:=((-w15*s1^2*s2^2-6*w15*s1^2*s2+12*w15*s1^2+w15*s2^2+6*w15*s2-12*w15+5*w3*s1^2*s2^2+20*w3*s1^2*s2
-60*w3*s1^2-5*w3*s2^2-20*w3*s2+60*w3+3*w5*s1^2*s2^2-2*w5*s1^2*s2-36*w5*s1^2-6*w5*s1*s2^2
+4*w5*s1*s2+72*w5*s1+3*w5*s2^2-2*w5*s2-36*w5-5*s1^2*s2^2+60*s1^2+10*s1*s2^2-120*s1-5*s2^2
+60)*s3+(w15*s1^2*s2^2+w15*s1^2*s2-12*w15*s1^2-2*w15*s1*s2^2-2*w15*s1*s2+24*w15*s1+w15*s2^2
+w15*s2-12*w15-5*w3*s1^2*s2+5*w3*s2-w5*s1^2*s2^2-w5*s1^2*s2+12*w5*s1^2+2*w5*s1*s2^2+2*w5*s1*s2
-24*w5*s1-w5*s2^2-w5*s2+12*w5+15*s1^2*s2-15*s2))*I+(-3*w15*s1^2*s2^2+2*w15*s1^2*s2+36*w15*s1^2
+3*w15*s2^2-2*w15*s2-36*w15+5*w3*s1^2*s2^2-60*w3*s1^2-5*w3*s2^2+60*w3-w5*s1^2*s2^2-6*w5*s1^2*s2
+12*w5*s1^2+2*w5*s1*s2^2+12*w5*s1*s2-24*w5*s1-w5*s2^2-6*w5*s2+12*w5+5*s1^2*s2^2+20*s1^2*s2
-60*s1^2-10*s1*s2^2-40*s1*s2+120*s1+5*s2^2+20*s2-60)*s3+w15*s1^2*s2^2+w15*s1^2*s2-12*w15*s1^2
-w15*s2^2-w15*s2+12*w15+5*w3*s1^2*s2-10*w3*s1*s2+5*w3*s2-3*w5*s1^2*s2^2-3*w5*s1^2*s2+36*w5*s1^2
+3*w5*s2^2+3*w5*s2-36*w5-5*s1^2*s2+10*s1*s2-5*s2; c07:=120*w3*s1*I-120*s1;
c08:=((-30*w15+90*w3-30*w5-30)*s3+(30*w5-90))*I+(-30*w15+90*w3+30*w5+30)*s3+30*w15+30*w3;
c09:=((w15*s1^2*s2^2+6*w15*s1^2*s2-12*w15*s1^2-3*w15*s1*s2^2-3*w15*s1*s2+36*w15*s1+2*w15*s2^2-3*w15*s2
-24*w15-5*w3*s1^2*s2^2-20*w3*s1^2*s2+60*w3*s1^2+5*w3*s1*s2^2+5*w3*s1*s2-60*w3*s1+15*w3*s2
91
APPENDIX A.
-3*w5*s1^2*s2^2-8*w5*s1^2*s2+36*w5*s1^2+6*w5*s1*s2^2+w5*s1*s2-72*w5*s1-3*w5*s2^2+7*w5*s2+36*w5
+5*s1^2*s2^2+30*s1^2*s2-60*s1^2-10*s1*s2^2-15*s1*s2+120*s1+5*s2^2-15*s2-60)*s3-w15*s1^2*s2^2
+4*w15*s1^2*s2+12*w15*s1^2+1/2*w15*s1*s2^2-2*w15*s1*s2-6*w15*s1+1/2*w15*s2^2-2*w15*s2-6*w15
+10*w3*s1^2*s2+5/2*w3*s1*s2^2-30*w3*s1-5/2*w3*s2^2-10*w3*s2+30*w3+w5*s1^2*s2^2-4*w5*s1^2*s2
-12*w5*s1^2-7/2*w5*s1*s2^2-w5*s1*s2+42*w5*s1+5/2*w5*s2^2+5*w5*s2-30*w5-15/2*s1*s2^2-15*s1*s2
+90*s1+15/2*s2^2+15*s2-90)*I+(3*w15*s1^2*s2^2-2*w15*s1^2*s2-36*w15*s1^2+5*w15*s1*s2-3*w15*s2^2
-3*w15*s2+36*w15-5*w3*s1^2*s2^2+60*w3*s1^2-15*w3*s1*s2+5*w3*s2^2+15*w3*s2-60*w3-5*w5*s1^2*s2^2
+60*w5*s1^2+w5*s1*s2^2-9*w5*s1*s2-12*w5*s1+4*w5*s2^2+9*w5*s2-48*w5+5*s1^2*s2^2-10*s1^2*s2
-60*s1^2+5*s1*s2^2+35*s1*s2-60*s1-10*s2^2-25*s2+120)*s3-2*w15*s1^2*s2^2-2*w15*s1^2*s2
+24*w15*s1^2+1/2*w15*s1*s2^2+3*w15*s1*s2-6*w15*s1+3/2*w15*s2^2-w15*s2-18*w15-5*w3*s1^2*s2^2
-10*w3*s1^2*s2+60*w3*s1^2+5/2*w3*s1*s2^2+5*w3*s1*s2-30*w3*s1+5/2*w3*s2^2+5*w3*s2-30*w3
+3/2*w5*s1*s2^2-6*w5*s1*s2-18*w5*s1-3/2*w5*s2^2+6*w5*s2+18*w5+5*s1^2*s2^2+10*s1^2*s2-60*s1^2
-5/2*s1*s2^2-20*s1*s2+30*s1-5/2*s2^2+10*s2+30;
phi:=[ c01, c02, c03, c04, c05, c06, c07, c08, c09 ];
Although the simpler fiducial vector given by Appleby et al. [4] is simple enough to be
derived by hand, the calculations still occupy 6 pages of the paper.
A.3
d = 12 Fiducial Vector [3]
The following is a fiducial vector given by Appleby et al. [3] using the k-nomial representation of the Clifford group. It is considerably simpler than the fiducial vector given by Grassl
[28], the listing of which occupies 6 pages.
|ψ0 i =
where
and
q√
s1 = ( 13 − 1)/2,
11
X
j=0
xi |bi i
q √
s2 = (3 13 + 9)/2,
t31 = 12t1 + 10,
√
√
√
√
√
x0 = ((−30 13 − 312)s1 + (24 13t21 − 102 13t1 − 309 13 − 92t21 − 158t1 − 5)) −1
√
√
√
√
√
√
√
√
+ (26 39t21 − 364 3)s1 + 28 39t21 − 58 39t1 − 147 39 + 96 3t21 − 42 3t1 − 443 3
√
√
√
√
x1 = (24 13t21 − 102 13t1 − 540 13 − 92t21 − 158t1 − 980) −1
√
√
√
√
√
√
+ 28 39t21 − 58 39t1 − 264 39 + 96 3t21 − 42 3t1 − 1184 3
√
√
√
√
√
x2 = ((24 13 − 702)s1 − 54 13t21 + 138 13t1 + 375 13 − 98t21 + 142t1 + 667) −1
√
√
√
√
√
√
√
√
+ (28 39 − 26 3)s1 − 2 39t21 − 22 39t1 − 81 39 − 94 3t21 − 58 3t1 + 219 3
APPENDIX A.
92
√
√
√
√
√
x3 = ((−30 13 − 312)s1 − 54 13t21 + 138 13t1 + 315 13 − 98t21 + 142t1 + 43) −1
√
√
√
√
√
√
√
√
+ (26 39 − 364 3)s1 − 2 39t21 − 22 39t1 + 93 39 − 94 3t21 − 58 3t1 + 1077 3
√
√
√
√
x4 = (30 13t21 − 36 13t1 − 588 13 + 190t21 + 16t1 − 3236) −1
√
√
√
√
√
√
− 26 39t21 + 80 39t1 + 168 39 − 2 3t21 + 100 3t1 − 400 3
√
√
√
√
√
x5 = ((24 13 − 702)s1 + (30 13t21 − 36 13t1 − 297 13 + 190t21 + 16t1 − 1637)) −1
√
√
√
√
√
√
√
√
+ (28 39 − 26 3)s1 − 26 39t21 + 80 39t1 + 111 39 − 2 3t21 + 100 3t1 − 517 3
√
√
√
√
√
x6 = ((−30 13 − 312)s1 + (30 13t21 − 36 13t1 − 357 13 + 190t21 + 16t1 − 2261)) −1
√
√
√
√
√
√
√
√
+ (26 39 − 364 3)s1 − 26 39t21 + 80 39t1 + 285 39 − 2 3t21 + 100 3t1 + 341 3
√
x7 = 488 13s2
√
√
√
√
√
x8 = ((24 13 − 702)s1 + (24 13t21 − 102 13t1 − 249 13 − 92t21 − 158t1 + 619)) −1
√
√
√
√
√
√
√
√
+ (28 39 − 26 3)s1 + 28 39t21 − 58 39t1 − 321 39 + 96 3t21 − 42 3t1 − 1301 3
√
√
√
√
x9 = ((85 39 + 91 3)s1 s2 − 122 39s2 ) −1
√
√
+ (23 13 − 871)s1 s2 + 122 13s2
√
√
√
√
x10 = (−54 13t21 + 138 13t1 + 84 13 − 98t21 + 142t1 − 932) −1
√
√
√
√
√
√
− 2 39t21 − 22 39t1 − 24 39 − 94 3t21 − 58 3t1 + 336 3
√
√
√
√
x11 = ((31 39 + 481 3)s1 s2 + 122 39s2 ) −1
√
√
+ (139 13 − 299)s1 s2 + 122 13s2
A.4
d = 16 Fiducial Vector [4]
The following fiducial vector is the first exact fiducial for d = 16 as given by Appleby et al.
[4].
w2:=Sqrt(2);
w13:=-Sqrt(13);
w17:=-Sqrt(17);
I:=Sqrt(-1);
w26:=w2*w13;
w34:=w2*w17;
w221:=w13*w17;
w442:=w2*w13*w17;
r2:=Sqrt(w13*w17-11);
APPENDIX A.
r3:=Sqrt(15+w17);
t1:=Sqrt(15+(4-w17)*r3-3*w17);
t2:=Sqrt((((-5*w17+3)*w13+(39*w17-65))*r3+((16*w17-72)*w13+936))*t1-208*w13+2288);
t3:=Sqrt(2-w2); t4:=Sqrt(2+t3);
c01:=(-20/13*w26*r3*t1*t2+((-90*w2+60*w13-20*w17+30*w26+10*w34+20*w221+10*w442-60)*r3
+(60*w13+60*w17+20*w221-140))*t1*t3+((350*w2+160*w13-40*w17+150*w26-50*w34-10*w442+760)*r3
+(400*w2+440*w13-600*w17+320*w26-320*w34-40*w221-80*w442+360))*t3)*I+(-40/13*w13
-20/13*w26)*r3*t1*t2+((140*w2+20*w26+160)*r3-20*w2-260*w13-500*w17-140*w26-140*w34-60*w221
-20*w442-1260)*t1*t3+((530*w2-60*w13-100*w17-30*w26-70*w34+20*w221+10*w442+620)*r3+(520*w2
-400*w13-1120*w17-200*w26-520*w34+40*w442-1200))*t3;
c02:=(((10*w2-4*w26-2*w34)*r2*r3+(-24*w2-24*w26-12*w34-4*w442)*r2+(-50*w13+10*w17-10*w221
-30)*r3-20*w13-140*w17-20*w221-540)*t1*t3*t4+((-53/2*w2+w13+3*w17-7/2*w26-1/2*w34-w221
-3/2*w442-11)*r2*r3+(-75*w2+26*w13-2*w17+5*w26-5*w34+2*w221-5*w442+22)*r2+(25*w2+25*w13
+5*w17+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221
-10*w442-10)*t1*t3+((-10*w2-27*w13-21*w17+4*w26+2*w34-3*w221-53)*r2*r3+(24*w2-24*w13
+8*w17+24*w26+12*w34-8*w221+4*w442-88)*r2+(-85*w2+50*w13-10*w17-25*w26+15*w34+10*w221
-5*w442+30)*r3+(80*w2+20*w13+140*w17+40*w26+40*w34+20*w221+540))*t1*t4+((53/2*w2+11*w13
+3*w17+27/2*w26+21/2*w34+3*w221+3/2*w442+43)*r2*r3+(44*w2+14*w13+22*w17+12*w26-4*w34
+14*w221+4*w442+174)*r2+(10*w2-25*w13+15*w17+10*w34-5*w221-85)*r3-280*w2+40*w13+40*w17
-80*w26-140*w34-20*w442+80)*t1+((-36*w2-28*w26-4*w34+4*w442)*r2*r3+(122*w2-86*w26+22*w34
+22*w442)*r2+(-80*w13+40*w17-600)*r3-220*w13+260*w17-20*w221-1300)*t3*t4+((-46*w2+25*w13
+15*w17+24*w26+12*w34-3*w221-6*w442-13)*r2*r3+(-394*w2+66*w13-2*w17+46*w26-2*w34-18*w221
-34*w442-238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((36*w2-56*w13-28*w17+28*w26+4*w34-4*w221-4*w442
-144)*r2*r3+(-122*w2-64*w13+48*w17+86*w26-22*w34+16*w221-22*w442+96)*r2+(-470*w2+80*w13
-40*w17+10*w26+30*w34-10*w442+600)*r3-240*w2+220*w13-260*w17+40*w26+400*w34+20*w221
-40*w442+1300)*t4+((72*w2-20*w13-20*w17+28*w26+14*w34+8*w221+2*w442+128)*r2*r3+(-48*w2
-6*w13-18*w17+32*w26-24*w34+46*w221-8*w442+666)*r2+(140*w2+10*w13+30*w17+20*w26-40*w34
-10*w221-470)*r3-190*w2+40*w13+400*w17-50*w26-290*w34-40*w221-30*w442-240))*I+((-3*w13
-9*w17+w221+11)*r2*r3+(56*w13+48*w17+24*w221+304)*r2+(-55*w2+5*w26-15*w34+5*w442)*r3
-60*w2+100*w26+100*w34+20*w442)*t1*t3*t4+((95/2*w2+11*w13+13*w17+37/2*w26+31/2*w34+w221
+5/2*w442+31)*r2*r3+(-33*w2-40*w13-40*w17-21*w26-23*w34-12*w221-3*w442-132)*r2+(55*w2
-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17-80*w26-140*w34
-10*w221-20*w442-10)*t1*t3+((-32*w2+3*w13+9*w17-14*w26-12*w34-w221-2*w442-11)*r2*r3
+(130*w2-56*w13-48*w17+42*w26+26*w34-24*w221+10*w442-304)*r2+(55*w2+10*w13-10*w17-5*w26
+15*w34+10*w221-5*w442-90)*r3+(60*w2+40*w13-80*w17-100*w26-100*w34-20*w442-680))*t1*t4
+((-73/2*w2+14*w13+12*w17-19/2*w26-17/2*w34+2*w221-3/2*w442+32)*r2*r3+(-20*w2-42*w13
-26*w17+12*w26+16*w34-10*w221-130)*r2+(-45*w2-25*w13-5*w17+5*w26-5*w34-5*w221+5*w442
+55)*r3-340*w2-60*w13-180*w17+20*w26-40*w34-20*w221-620)*t1+((-46*w13-38*w17+6*w221
+86)*r2*r3+(56*w13+8*w17+72*w221+952)*r2+(-60*w2+20*w26+40*w34)*r3-280*w2+160*w26
+120*w34)*t3*t4+((75*w2+23*w13+29*w17+27*w26+21*w34+3*w221+5*w442-7)*r2*r3+(-130*w2
-82*w13-6*w17-42*w26-26*w34-22*w221-10*w442-402)*r2+(-35*w2-90*w13-10*w17-55*w26-15*w34
+10*w221+5*w442+90)*r3-390*w2-260*w13-100*w17-210*w26-130*w34+60*w221+10*w442+60)*t3
+((-42*w2+46*w13+38*w17-26*w26-18*w34-6*w221-2*w442-86)*r2*r3+(286*w2-56*w13-8*w17+62*w26
+26*w34-72*w221+26*w442-952)*r2+(60*w2+80*w13-20*w26-40*w34+160)*r3+(280*w2+220*w13
93
APPENDIX A.
-340*w17-160*w26-120*w34-60*w221-940))*t4+(-36*w2+26*w13+18*w17-10*w34+2*w221-6*w442
+42)*r2*r3+(-74*w2-62*w13-26*w17+54*w26+2*w34-26*w221-14*w442-286)*r2+(80*w2-90*w13
+10*w17+40*w26+10*w221-130)*r3-470*w2-260*w13-140*w17+110*w26-170*w34+20*w221-30*w442
-1060;
c03:=(((-11/26*w13-1/2*w17-3/26*w221-1/2)*r2*r3+(10*w2+20/13*w26+10/13*w442)*r2
-40/13*w13*r3)*t1*t2+((-43*w2-27*w13-21*w17-11*w26-3*w34-3*w221-3*w442-53)*r2*r3+(-174*w2
-24*w13+8*w17-14*w26-22*w34-8*w221-14*w442-88)*r2+(-55*w2-10*w13+10*w17+25*w26+5*w34
-10*w221+5*w442+90)*r3+(620*w2-40*w13+80*w17+60*w26+180*w34+20*w442+680))*t1+((-128*w2
-56*w13-28*w17+20*w26+20*w34-4*w221-8*w442-144)*r2*r3+(-666*w2-64*w13+48*w17+6*w26+18*w34
+16*w221-46*w442+96)*r2+(130*w2-80*w13+90*w26-10*w34-10*w442-160)*r3+(1060*w2-220*w13
+340*w17+260*w26+140*w34+60*w221-20*w442+940)))*I+((11/26*w13+1/2*w17+3/26*w221+1/2)*r2*r3
+(10*w2+20/13*w26+10/13*w442)*r2)*t1*t2+((-32*w2+19*w13+17*w17-14*w26-12*w34+3*w221-2*w442
+73)*r2*r3+(130*w2-24*w13-32*w17+42*w26+26*w34+10*w442+40)*r2+(-85*w2+20*w17-25*w26
+15*w34-5*w442+20)*r3+(80*w2-160*w13-280*w17+40*w26+40*w34-40*w221-560))*t1+(-42*w2
+20*w17-26*w26-18*w34+12*w221-2*w442+72)*r2*r3+(286*w2-108*w13-4*w17+62*w26+26*w34
+28*w221+26*w442+148)*r2+(-470*w2+40*w13-80*w17+10*w26+30*w34-10*w442+280)*r3-240*w2
-100*w13-580*w17+40*w26+400*w34-60*w221-40*w442-380;
c04:=(((-63*w2-23*w26-19*w34-3*w442)*r2*r3+(-64*w2+20*w34-4*w442)*r2+(-20*w17+140)*r3
-100*w13-220*w17-20*w221-700)*t1*t3*t4+((53/2*w2-w13-3*w17+7/2*w26+1/2*w34+w221+3/2*w442
+11)*r2*r3+(75*w2-26*w13+2*w17-5*w26+5*w34-2*w221+5*w442-22)*r2+(25*w2+25*w13+5*w17
+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221-10*w442
-10)*t1*t3+((63*w2-19*w13-17*w17+23*w26+19*w34-3*w221+3*w442-73)*r2*r3+(64*w2+24*w13
+32*w17-20*w34+4*w442-40)*r2+(55*w2+20*w17-25*w26-5*w34-5*w442-140)*r3-620*w2+100*w13
+220*w17-60*w26-180*w34+20*w221-20*w442+700)*t1*t4+((53/2*w2+11*w13+3*w17+27/2*w26
+21/2*w34+3*w221+3/2*w442+43)*r2*r3+(44*w2+14*w13+22*w17+12*w26-4*w34+14*w221+4*w442
+174)*r2+(-10*w2+25*w13-15*w17-10*w34+5*w221+85)*r3+(280*w2-40*w13-40*w17+80*w26+140*w34
+20*w442-80))*t1+((-108*w2-28*w26-24*w34-8*w442)*r2*r3+(-26*w2+22*w26+26*w34-6*w442)*r2
+(-100*w13-20*w17+20*w221+340)*r3-300*w13-540*w17+60*w221-820)*t3*t4+((46*w2-25*w13
-15*w17-24*w26-12*w34+3*w221+6*w442+13)*r2*r3+(394*w2-66*w13+2*w17-46*w26+2*w34+18*w221
+34*w442+238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((108*w2-20*w17+28*w26+24*w34-12*w221+8*w442
-72)*r2*r3+(26*w2+108*w13+4*w17-22*w26-26*w34-28*w221+6*w442-148)*r2+(-130*w2+100*w13
+20*w17-90*w26+10*w34-20*w221+10*w442-340)*r3-1060*w2+300*w13+540*w17-260*w26-140*w34
-60*w221+20*w442+820)*t4+((72*w2-20*w13-20*w17+28*w26+14*w34+8*w221+2*w442+128)*r2*r3
+(-48*w2-6*w13-18*w17+32*w26-24*w34+46*w221-8*w442+666)*r2+(-140*w2-10*w13-30*w17-20*w26
+40*w34+10*w221+470)*r3+(190*w2-40*w13-400*w17+50*w26+290*w34+40*w221+30*w442+240)))*I
+((-25*w13-15*w17-5*w221-75)*r2*r3+(28*w13+4*w17-4*w221-44)*r2+(35*w2-5*w26-5*w34
-5*w442)*r3+(620*w2+60*w26+180*w34+20*w442))*t1*t3*t4+((-95/2*w2-11*w13-13*w17-37/2*w26
-31/2*w34-w221-5/2*w442-31)*r2*r3+(33*w2+40*w13+40*w17+21*w26+23*w34+12*w221+3*w442
+132)*r2+(55*w2-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17
-80*w26-140*w34-10*w221-20*w442-10)*t1*t3+((-43*w2+25*w13+15*w17-11*w26-3*w34+5*w221
-3*w442+75)*r2*r3+(-174*w2-28*w13-4*w17-14*w26-22*w34+4*w221-14*w442+44)*r2+(-35*w2
-20*w17+5*w26+5*w34+5*w442-20)*r3-620*w2+160*w13+280*w17-60*w26-180*w34+40*w221-20*w442
+560)*t1*t4+((-73/2*w2+14*w13+12*w17-19/2*w26-17/2*w34+2*w221-3/2*w442+32)*r2*r3+(-20*w2
-42*w13-26*w17+12*w26+16*w34-10*w221-130)*r2+(45*w2+25*w13+5*w17-5*w26+5*w34+5*w221
-5*w442-55)*r3+(340*w2+60*w13+180*w17-20*w26+40*w34+20*w221+620))*t1+((-6*w13+2*w17
-10*w221-170)*r2*r3+(68*w13+44*w17-20*w221-380)*r2+(-220*w2-60*w26+40*w34)*r3+(660*w2
94
APPENDIX A.
-60*w26+460*w34+60*w442))*t3*t4+((-75*w2-23*w13-29*w17-27*w26-21*w34-3*w221-5*w442
+7)*r2*r3+(130*w2+82*w13+6*w17+42*w26+26*w34+22*w221+10*w442+402)*r2+(-35*w2-90*w13
-10*w17-55*w26-15*w34+10*w221+5*w442+90)*r3-390*w2-260*w13-100*w17-210*w26-130*w34
+60*w221+10*w442+60)*t3+((-128*w2+6*w13-2*w17+20*w26+20*w34+10*w221-8*w442+170)*r2*r3
+(-666*w2-68*w13-44*w17+6*w26+18*w34+20*w221-46*w442+380)*r2+(220*w2-40*w13+80*w17+60*w26
-40*w34-280)*r3-660*w2+100*w13+580*w17+60*w26-460*w34+60*w221-60*w442+380)*t4+(-36*w2
+26*w13+18*w17-10*w34+2*w221-6*w442+42)*r2*r3+(-74*w2-62*w13-26*w17+54*w26+2*w34-26*w221
-14*w442-286)*r2+(-80*w2+90*w13-10*w17-40*w26-10*w221+130)*r3+470*w2+260*w13+140*w17
-110*w26+170*w34-20*w221+30*w442+1060;
c05:=(-20/13*w26*r3*t1*t2+((20*w2+30*w13+30*w17+12*w26+16*w34+2*w221+42)*r2*r3+(-110*w2
-32*w13-56*w17-14*w26-42*w34-16*w221-10*w442-216)*r2)*t1*t3+((-20*w2+102*w13+66*w17+48*w26
+44*w34-2*w221+58)*r2*r3+(-640*w2+8*w13-56*w17-16*w26-8*w34-88*w221-40*w442
-1048)*r2)*t3)*I+(40/13*w13-20/13*w26)*r3*t1*t2+((42*w2+44*w13+32*w17+10*w26+10*w34+8*w221
+2*w442+148)*r2*r3+(-154*w2-52*w13-36*w17-66*w26-38*w34+4*w221-14*w442+84)*r2)*t1*t3
+((6*w2+6*w13+18*w17-2*w26+14*w34+22*w221+6*w442+242)*r2*r3+(-164*w2-176*w13-48*w17
-148*w26-4*w34+48*w221-4*w442+528)*r2)*t3;
c06:=(((-63*w2-23*w26-19*w34-3*w442)*r2*r3+(-64*w2+20*w34-4*w442)*r2+(20*w17-140)*r3
+(100*w13+220*w17+20*w221+700))*t1*t3*t4+((-53/2*w2+w13+3*w17-7/2*w26-1/2*w34-w221
-3/2*w442-11)*r2*r3+(-75*w2+26*w13-2*w17+5*w26-5*w34+2*w221-5*w442+22)*r2+(25*w2+25*w13
+5*w17+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221
-10*w442-10)*t1*t3+((63*w2-19*w13-17*w17+23*w26+19*w34-3*w221+3*w442-73)*r2*r3+(64*w2
+24*w13+32*w17-20*w34+4*w442-40)*r2+(-55*w2-20*w17+25*w26+5*w34+5*w442+140)*r3+(620*w2
-100*w13-220*w17+60*w26+180*w34-20*w221+20*w442-700))*t1*t4+((-53/2*w2-11*w13-3*w17
-27/2*w26-21/2*w34-3*w221-3/2*w442-43)*r2*r3+(-44*w2-14*w13-22*w17-12*w26+4*w34-14*w221
-4*w442-174)*r2+(-10*w2+25*w13-15*w17-10*w34+5*w221+85)*r3+(280*w2-40*w13-40*w17+80*w26
+140*w34+20*w442-80))*t1+((-108*w2-28*w26-24*w34-8*w442)*r2*r3+(-26*w2+22*w26+26*w34
-6*w442)*r2+(100*w13+20*w17-20*w221-340)*r3+(300*w13+540*w17-60*w221+820))*t3*t4+((-46*w2
+25*w13+15*w17+24*w26+12*w34-3*w221-6*w442-13)*r2*r3+(-394*w2+66*w13-2*w17+46*w26-2*w34
-18*w221-34*w442-238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13
-420*w17+10*w26-430*w34-20*w221-10*w442+460)*t3+((108*w2-20*w17+28*w26+24*w34-12*w221
+8*w442-72)*r2*r3+(26*w2+108*w13+4*w17-22*w26-26*w34-28*w221+6*w442-148)*r2+(130*w2
-100*w13-20*w17+90*w26-10*w34+20*w221-10*w442+340)*r3+(1060*w2-300*w13-540*w17+260*w26
+140*w34+60*w221-20*w442-820))*t4+((-72*w2+20*w13+20*w17-28*w26-14*w34-8*w221-2*w442
-128)*r2*r3+(48*w2+6*w13+18*w17-32*w26+24*w34-46*w221+8*w442-666)*r2+(-140*w2-10*w13
-30*w17-20*w26+40*w34+10*w221+470)*r3+(190*w2-40*w13-400*w17+50*w26+290*w34+40*w221
+30*w442+240)))*I+((-25*w13-15*w17-5*w221-75)*r2*r3+(28*w13+4*w17-4*w221-44)*r2+(-35*w2
+5*w26+5*w34+5*w442)*r3-620*w2-60*w26-180*w34-20*w442)*t1*t3*t4+((95/2*w2+11*w13+13*w17
+37/2*w26+31/2*w34+w221+5/2*w442+31)*r2*r3+(-33*w2-40*w13-40*w17-21*w26-23*w34-12*w221
-3*w442-132)*r2+(55*w2-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13
-70*w17-80*w26-140*w34-10*w221-20*w442-10)*t1*t3+((-43*w2+25*w13+15*w17-11*w26-3*w34
+5*w221-3*w442+75)*r2*r3+(-174*w2-28*w13-4*w17-14*w26-22*w34+4*w221-14*w442+44)*r2+(35*w2
+20*w17-5*w26-5*w34-5*w442+20)*r3+(620*w2-160*w13-280*w17+60*w26+180*w34-40*w221+20*w442
-560))*t1*t4+((73/2*w2-14*w13-12*w17+19/2*w26+17/2*w34-2*w221+3/2*w442-32)*r2*r3+(20*w2
+42*w13+26*w17-12*w26-16*w34+10*w221+130)*r2+(45*w2+25*w13+5*w17-5*w26+5*w34+5*w221
-5*w442-55)*r3+(340*w2+60*w13+180*w17-20*w26+40*w34+20*w221+620))*t1+((-6*w13+2*w17
-10*w221-170)*r2*r3+(68*w13+44*w17-20*w221-380)*r2+(220*w2+60*w26-40*w34)*r3-660*w2+60*w26
-460*w34-60*w442)*t3*t4+((75*w2+23*w13+29*w17+27*w26+21*w34+3*w221+5*w442-7)*r2*r3
95
APPENDIX A.
+(-130*w2-82*w13-6*w17-42*w26-26*w34-22*w221-10*w442-402)*r2+(-35*w2-90*w13-10*w17-55*w26
-15*w34+10*w221+5*w442+90)*r3-390*w2-260*w13-100*w17-210*w26-130*w34+60*w221+10*w442
+60)*t3+((-128*w2+6*w13-2*w17+20*w26+20*w34+10*w221-8*w442+170)*r2*r3+(-666*w2-68*w13
-44*w17+6*w26+18*w34+20*w221-46*w442+380)*r2+(-220*w2+40*w13-80*w17-60*w26+40*w34+280)*r3
+(660*w2-100*w13-580*w17-60*w26+460*w34-60*w221+60*w442-380))*t4+(36*w2-26*w13-18*w17
+10*w34-2*w221+6*w442-42)*r2*r3+(74*w2+62*w13+26*w17-54*w26-2*w34+26*w221+14*w442+286)*r2
+(-80*w2+90*w13-10*w17-40*w26-10*w221+130)*r3+470*w2+260*w13+140*w17-110*w26+170*w34
-20*w221+30*w442+1060;
c07:=(((11/26*w13+1/2*w17+3/26*w221+1/2)*r2*r3+(-10*w2-20/13*w26-10/13*w442)*r2
-40/13*w13*r3)*t1*t2+((-43*w2-27*w13-21*w17-11*w26-3*w34-3*w221-3*w442-53)*r2*r3+(-174*w2
-24*w13+8*w17-14*w26-22*w34-8*w221-14*w442-88)*r2+(55*w2+10*w13-10*w17-25*w26-5*w34
+10*w221-5*w442-90)*r3-620*w2+40*w13-80*w17-60*w26-180*w34-20*w442-680)*t1+((-128*w2
-56*w13-28*w17+20*w26+20*w34-4*w221-8*w442-144)*r2*r3+(-666*w2-64*w13+48*w17+6*w26+18*w34
+16*w221-46*w442+96)*r2+(-130*w2+80*w13-90*w26+10*w34+10*w442+160)*r3-1060*w2+220*w13
-340*w17-260*w26-140*w34-60*w221+20*w442-940))*I+((-11/26*w13-1/2*w17-3/26*w221-1/2)*r2*r3
+(-10*w2-20/13*w26-10/13*w442)*r2)*t1*t2+((-32*w2+19*w13+17*w17-14*w26-12*w34+3*w221-2*w442
+73)*r2*r3+(130*w2-24*w13-32*w17+42*w26+26*w34+10*w442+40)*r2+(85*w2-20*w17+25*w26-15*w34
+5*w442-20)*r3-80*w2+160*w13+280*w17-40*w26-40*w34+40*w221+560)*t1+(-42*w2+20*w17-26*w26
-18*w34+12*w221-2*w442+72)*r2*r3+(286*w2-108*w13-4*w17+62*w26+26*w34+28*w221+26*w442
+148)*r2+(470*w2-40*w13+80*w17-10*w26-30*w34+10*w442-280)*r3+240*w2+100*w13+580*w17
-40*w26-400*w34+60*w221+40*w442+380;
c08:=(((10*w2-4*w26-2*w34)*r2*r3+(-24*w2-24*w26-12*w34-4*w442)*r2+(50*w13-10*w17+10*w221
+30)*r3+(20*w13+140*w17+20*w221+540))*t1*t3*t4+((53/2*w2-w13-3*w17+7/2*w26+1/2*w34+w221
+3/2*w442+11)*r2*r3+(75*w2-26*w13+2*w17-5*w26+5*w34-2*w221+5*w442-22)*r2+(25*w2+25*w13
+5*w17+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221
-10*w442-10)*t1*t3+((-10*w2-27*w13-21*w17+4*w26+2*w34-3*w221-53)*r2*r3+(24*w2-24*w13
+8*w17+24*w26+12*w34-8*w221+4*w442-88)*r2+(85*w2-50*w13+10*w17+25*w26-15*w34-10*w221
+5*w442-30)*r3-80*w2-20*w13-140*w17-40*w26-40*w34-20*w221-540)*t1*t4+((-53/2*w2-11*w13
-3*w17-27/2*w26-21/2*w34-3*w221-3/2*w442-43)*r2*r3+(-44*w2-14*w13-22*w17-12*w26+4*w34
-14*w221-4*w442-174)*r2+(10*w2-25*w13+15*w17+10*w34-5*w221-85)*r3-280*w2+40*w13+40*w17
-80*w26-140*w34-20*w442+80)*t1+((-36*w2-28*w26-4*w34+4*w442)*r2*r3+(122*w2-86*w26+22*w34
+22*w442)*r2+(80*w13-40*w17+600)*r3+(220*w13-260*w17+20*w221+1300))*t3*t4+((46*w2-25*w13
-15*w17-24*w26-12*w34+3*w221+6*w442+13)*r2*r3+(394*w2-66*w13+2*w17-46*w26+2*w34+18*w221
+34*w442+238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((36*w2-56*w13-28*w17+28*w26+4*w34-4*w221-4*w442
-144)*r2*r3+(-122*w2-64*w13+48*w17+86*w26-22*w34+16*w221-22*w442+96)*r2+(470*w2-80*w13
+40*w17-10*w26-30*w34+10*w442-600)*r3+(240*w2-220*w13+260*w17-40*w26-400*w34-20*w221
+40*w442-1300))*t4+((-72*w2+20*w13+20*w17-28*w26-14*w34-8*w221-2*w442-128)*r2*r3+(48*w2
+6*w13+18*w17-32*w26+24*w34-46*w221+8*w442-666)*r2+(140*w2+10*w13+30*w17+20*w26-40*w34
-10*w221-470)*r3-190*w2+40*w13+400*w17-50*w26-290*w34-40*w221-30*w442-240))*I+((-3*w13
-9*w17+w221+11)*r2*r3+(56*w13+48*w17+24*w221+304)*r2+(55*w2-5*w26+15*w34-5*w442)*r3
+(60*w2-100*w26-100*w34-20*w442))*t1*t3*t4+((-95/2*w2-11*w13-13*w17-37/2*w26-31/2*w34-w221
-5/2*w442-31)*r2*r3+(33*w2+40*w13+40*w17+21*w26+23*w34+12*w221+3*w442+132)*r2+(55*w2
-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17-80*w26-140*w34
-10*w221-20*w442-10)*t1*t3+((-32*w2+3*w13+9*w17-14*w26-12*w34-w221-2*w442-11)*r2*r3
+(130*w2-56*w13-48*w17+42*w26+26*w34-24*w221+10*w442-304)*r2+(-55*w2-10*w13+10*w17+5*w26
-15*w34-10*w221+5*w442+90)*r3-60*w2-40*w13+80*w17+100*w26+100*w34+20*w442+680)*t1*t4
96
APPENDIX A.
+((73/2*w2-14*w13-12*w17+19/2*w26+17/2*w34-2*w221+3/2*w442-32)*r2*r3+(20*w2+42*w13+26*w17
-12*w26-16*w34+10*w221+130)*r2+(-45*w2-25*w13-5*w17+5*w26-5*w34-5*w221+5*w442+55)*r3
-340*w2-60*w13-180*w17+20*w26-40*w34-20*w221-620)*t1+((-46*w13-38*w17+6*w221+86)*r2*r3
+(56*w13+8*w17+72*w221+952)*r2+(60*w2-20*w26-40*w34)*r3+(280*w2-160*w26-120*w34))*t3*t4
+((-75*w2-23*w13-29*w17-27*w26-21*w34-3*w221-5*w442+7)*r2*r3+(130*w2+82*w13+6*w17+42*w26
+26*w34+22*w221+10*w442+402)*r2+(-35*w2-90*w13-10*w17-55*w26-15*w34+10*w221+5*w442+90)*r3
-390*w2-260*w13-100*w17-210*w26-130*w34+60*w221+10*w442+60)*t3+((-42*w2+46*w13+38*w17
-26*w26-18*w34-6*w221-2*w442-86)*r2*r3+(286*w2-56*w13-8*w17+62*w26+26*w34-72*w221+26*w442
-952)*r2+(-60*w2-80*w13+20*w26+40*w34-160)*r3-280*w2-220*w13+340*w17+160*w26+120*w34
+60*w221+940)*t4+(36*w2-26*w13-18*w17+10*w34-2*w221+6*w442-42)*r2*r3+(74*w2+62*w13
+26*w17-54*w26-2*w34+26*w221+14*w442+286)*r2+(80*w2-90*w13+10*w17+40*w26+10*w221-130)*r3
-470*w2-260*w13-140*w17+110*w26-170*w34+20*w221-30*w442-1060;
c09:=(-20/13*w26*r3*t1*t2+((90*w2-60*w13+20*w17-30*w26-10*w34-20*w221-10*w442+60)*r3-60*w13
-60*w17-20*w221+140)*t1*t3+((-350*w2-160*w13+40*w17-150*w26+50*w34+10*w442-760)*r3-400*w2
-440*w13+600*w17-320*w26+320*w34+40*w221+80*w442-360)*t3)*I+(-40/13*w13-20/13*w26)*r3*t1*t2
+((-140*w2-20*w26-160)*r3+(20*w2+260*w13+500*w17+140*w26+140*w34+60*w221+20*w442
+1260))*t1*t3+((-530*w2+60*w13+100*w17+30*w26+70*w34-20*w221-10*w442-620)*r3-520*w2
+400*w13+1120*w17+200*w26+520*w34-40*w442+1200)*t3;
c10:=(((-10*w2+4*w26+2*w34)*r2*r3+(24*w2+24*w26+12*w34+4*w442)*r2+(50*w13-10*w17+10*w221
+30)*r3+(20*w13+140*w17+20*w221+540))*t1*t3*t4+((-53/2*w2+w13+3*w17-7/2*w26-1/2*w34-w221
-3/2*w442-11)*r2*r3+(-75*w2+26*w13-2*w17+5*w26-5*w34+2*w221-5*w442+22)*r2+(25*w2+25*w13
+5*w17+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221
-10*w442-10)*t1*t3+((10*w2+27*w13+21*w17-4*w26-2*w34+3*w221+53)*r2*r3+(-24*w2+24*w13
-8*w17-24*w26-12*w34+8*w221-4*w442+88)*r2+(85*w2-50*w13+10*w17+25*w26-15*w34-10*w221
+5*w442-30)*r3-80*w2-20*w13-140*w17-40*w26-40*w34-20*w221-540)*t1*t4+((53/2*w2+11*w13
+3*w17+27/2*w26+21/2*w34+3*w221+3/2*w442+43)*r2*r3+(44*w2+14*w13+22*w17+12*w26-4*w34
+14*w221+4*w442+174)*r2+(10*w2-25*w13+15*w17+10*w34-5*w221-85)*r3-280*w2+40*w13+40*w17
-80*w26-140*w34-20*w442+80)*t1+((36*w2+28*w26+4*w34-4*w442)*r2*r3+(-122*w2+86*w26-22*w34
-22*w442)*r2+(80*w13-40*w17+600)*r3+(220*w13-260*w17+20*w221+1300))*t3*t4+((-46*w2+25*w13
+15*w17+24*w26+12*w34-3*w221-6*w442-13)*r2*r3+(-394*w2+66*w13-2*w17+46*w26-2*w34-18*w221
-34*w442-238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((-36*w2+56*w13+28*w17-28*w26-4*w34+4*w221+4*w442
+144)*r2*r3+(122*w2+64*w13-48*w17-86*w26+22*w34-16*w221+22*w442-96)*r2+(470*w2-80*w13
+40*w17-10*w26-30*w34+10*w442-600)*r3+(240*w2-220*w13+260*w17-40*w26-400*w34-20*w221
+40*w442-1300))*t4+((72*w2-20*w13-20*w17+28*w26+14*w34+8*w221+2*w442+128)*r2*r3+(-48*w2
-6*w13-18*w17+32*w26-24*w34+46*w221-8*w442+666)*r2+(140*w2+10*w13+30*w17+20*w26-40*w34
-10*w221-470)*r3-190*w2+40*w13+400*w17-50*w26-290*w34-40*w221-30*w442-240))*I+((3*w13
+9*w17-w221-11)*r2*r3+(-56*w13-48*w17-24*w221-304)*r2+(55*w2-5*w26+15*w34-5*w442)*r3
+(60*w2-100*w26-100*w34-20*w442))*t1*t3*t4+((95/2*w2+11*w13+13*w17+37/2*w26+31/2*w34+w221
+5/2*w442+31)*r2*r3+(-33*w2-40*w13-40*w17-21*w26-23*w34-12*w221-3*w442-132)*r2+(55*w2
-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17-80*w26-140*w34
-10*w221-20*w442-10)*t1*t3+((32*w2-3*w13-9*w17+14*w26+12*w34+w221+2*w442+11)*r2*r3
+(-130*w2+56*w13+48*w17-42*w26-26*w34+24*w221-10*w442+304)*r2+(-55*w2-10*w13+10*w17+5*w26
-15*w34-10*w221+5*w442+90)*r3-60*w2-40*w13+80*w17+100*w26+100*w34+20*w442+680)*t1*t4
+((-73/2*w2+14*w13+12*w17-19/2*w26-17/2*w34+2*w221-3/2*w442+32)*r2*r3+(-20*w2-42*w13
-26*w17+12*w26+16*w34-10*w221-130)*r2+(-45*w2-25*w13-5*w17+5*w26-5*w34-5*w221+5*w442
+55)*r3-340*w2-60*w13-180*w17+20*w26-40*w34-20*w221-620)*t1+((46*w13+38*w17-6*w221
97
APPENDIX A.
-86)*r2*r3+(-56*w13-8*w17-72*w221-952)*r2+(60*w2-20*w26-40*w34)*r3+(280*w2-160*w26
-120*w34))*t3*t4+((75*w2+23*w13+29*w17+27*w26+21*w34+3*w221+5*w442-7)*r2*r3+(-130*w2
-82*w13-6*w17-42*w26-26*w34-22*w221-10*w442-402)*r2+(-35*w2-90*w13-10*w17-55*w26-15*w34
+10*w221+5*w442+90)*r3-390*w2-260*w13-100*w17-210*w26-130*w34+60*w221+10*w442+60)*t3
+((42*w2-46*w13-38*w17+26*w26+18*w34+6*w221+2*w442+86)*r2*r3+(-286*w2+56*w13+8*w17-62*w26
-26*w34+72*w221-26*w442+952)*r2+(-60*w2-80*w13+20*w26+40*w34-160)*r3-280*w2-220*w13
+340*w17+160*w26+120*w34+60*w221+940)*t4+(-36*w2+26*w13+18*w17-10*w34+2*w221-6*w442
+42)*r2*r3+(-74*w2-62*w13-26*w17+54*w26+2*w34-26*w221-14*w442-286)*r2+(80*w2-90*w13
+10*w17+40*w26+10*w221-130)*r3-470*w2-260*w13-140*w17+110*w26-170*w34+20*w221-30*w442
-1060;
c11:=(((-11/26*w13-1/2*w17-3/26*w221-1/2)*r2*r3+(10*w2+20/13*w26+10/13*w442)*r2 40/13*w13*r3)*t1*t2+((43*w2+27*w13+21*w17+11*w26+3*w34+3*w221+3*w442+53)*r2*r3+(174*w2
+24*w13-8*w17+14*w26+22*w34+8*w221+14*w442+88)*r2+(55*w2+10*w13-10*w17-25*w26-5*w34
+10*w221-5*w442-90)*r3-620*w2+40*w13-80*w17-60*w26-180*w34-20*w442-680)*t1+((128*w2
+56*w13+28*w17-20*w26-20*w34+4*w221+8*w442+144)*r2*r3+(666*w2+64*w13-48*w17-6*w26-18*w34
-16*w221+46*w442-96)*r2+(-130*w2+80*w13-90*w26+10*w34+10*w442+160)*r3-1060*w2+220*w13
-340*w17-260*w26-140*w34-60*w221+20*w442-940))*I+((11/26*w13+1/2*w17+3/26*w221+1/2)*r2*r3
+(10*w2+20/13*w26+10/13*w442)*r2)*t1*t2+((32*w2-19*w13-17*w17+14*w26+12*w34-3*w221+2*w442
-73)*r2*r3+(-130*w2+24*w13+32*w17-42*w26-26*w34-10*w442-40)*r2+(85*w2-20*w17+25*w26
-15*w34+5*w442-20)*r3-80*w2+160*w13+280*w17-40*w26-40*w34+40*w221+560)*t1+(42*w2-20*w17
+26*w26+18*w34-12*w221+2*w442-72)*r2*r3+(-286*w2+108*w13+4*w17-62*w26-26*w34-28*w221
-26*w442-148)*r2+(470*w2-40*w13+80*w17-10*w26-30*w34+10*w442-280)*r3+240*w2+100*w13
+580*w17-40*w26-400*w34+60*w221+40*w442+380;
c12:=(((63*w2+23*w26+19*w34+3*w442)*r2*r3+(64*w2-20*w34+4*w442)*r2+(20*w17-140)*r3+(100*w13
+220*w17+20*w221+700))*t1*t3*t4+((53/2*w2-w13-3*w17+7/2*w26+1/2*w34+w221+3/2*w442
+11)*r2*r3+(75*w2-26*w13+2*w17-5*w26+5*w34-2*w221+5*w442-22)*r2+(25*w2+25*w13+5*w17
+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221-10*w442
-10)*t1*t3+((-63*w2+19*w13+17*w17-23*w26-19*w34+3*w221-3*w442+73)*r2*r3+(-64*w2-24*w13
-32*w17+20*w34-4*w442+40)*r2+(-55*w2-20*w17+25*w26+5*w34+5*w442+140)*r3+(620*w2-100*w13
-220*w17+60*w26+180*w34-20*w221+20*w442-700))*t1*t4+((53/2*w2+11*w13+3*w17+27/2*w26
+21/2*w34+3*w221+3/2*w442+43)*r2*r3+(44*w2+14*w13+22*w17+12*w26-4*w34+14*w221+4*w442
+174)*r2+(-10*w2+25*w13-15*w17-10*w34+5*w221+85)*r3+(280*w2-40*w13-40*w17+80*w26+140*w34
+20*w442-80))*t1+((108*w2+28*w26+24*w34+8*w442)*r2*r3+(26*w2-22*w26-26*w34+6*w442)*r2
+(100*w13+20*w17-20*w221-340)*r3+(300*w13+540*w17-60*w221+820))*t3*t4+((46*w2-25*w13
-15*w17-24*w26-12*w34+3*w221+6*w442+13)*r2*r3+(394*w2-66*w13+2*w17-46*w26+2*w34+18*w221
+34*w442+238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((-108*w2+20*w17-28*w26-24*w34+12*w221-8*w442
+72)*r2*r3+(-26*w2-108*w13-4*w17+22*w26+26*w34+28*w221-6*w442+148)*r2+(130*w2-100*w13
-20*w17+90*w26-10*w34+20*w221-10*w442+340)*r3+(1060*w2-300*w13-540*w17+260*w26+140*w34
+60*w221-20*w442-820))*t4+((72*w2-20*w13-20*w17+28*w26+14*w34+8*w221+2*w442+128)*r2*r3
+(-48*w2-6*w13-18*w17+32*w26-24*w34+46*w221-8*w442+666)*r2+(-140*w2-10*w13-30*w17-20*w26
+40*w34+10*w221+470)*r3+(190*w2-40*w13-400*w17+50*w26+290*w34+40*w221+30*w442+240)))*I
+((25*w13+15*w17+5*w221+75)*r2*r3+(-28*w13-4*w17+4*w221+44)*r2+(-35*w2+5*w26+5*w34
+5*w442)*r3-620*w2-60*w26-180*w34-20*w442)*t1*t3*t4+((-95/2*w2-11*w13-13*w17-37/2*w26
-31/2*w34-w221-5/2*w442-31)*r2*r3+(33*w2+40*w13+40*w17+21*w26+23*w34+12*w221+3*w442
+132)*r2+(55*w2-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17
-80*w26-140*w34-10*w221-20*w442-10)*t1*t3+((43*w2-25*w13-15*w17+11*w26+3*w34-5*w221
98
APPENDIX A.
+3*w442-75)*r2*r3+(174*w2+28*w13+4*w17+14*w26+22*w34-4*w221+14*w442-44)*r2+(35*w2+20*w17
-5*w26-5*w34-5*w442+20)*r3+(620*w2-160*w13-280*w17+60*w26+180*w34-40*w221+20*w442
-560))*t1*t4+((-73/2*w2+14*w13+12*w17-19/2*w26-17/2*w34+2*w221-3/2*w442+32)*r2*r3+(-20*w2
-42*w13-26*w17+12*w26+16*w34-10*w221-130)*r2+(45*w2+25*w13+5*w17-5*w26+5*w34+5*w221
-5*w442-55)*r3+(340*w2+60*w13+180*w17-20*w26+40*w34+20*w221+620))*t1+((6*w13-2*w17
+10*w221+170)*r2*r3+(-68*w13-44*w17+20*w221+380)*r2+(220*w2+60*w26-40*w34)*r3-660*w2
+60*w26-460*w34-60*w442)*t3*t4+((-75*w2-23*w13-29*w17-27*w26-21*w34-3*w221-5*w442
+7)*r2*r3+(130*w2+82*w13+6*w17+42*w26+26*w34+22*w221+10*w442+402)*r2+(-35*w2-90*w13
-10*w17-55*w26-15*w34+10*w221+5*w442+90)*r3-390*w2-260*w13-100*w17-210*w26-130*w34
+60*w221+10*w442+60)*t3+((128*w2-6*w13+2*w17-20*w26-20*w34-10*w221+8*w442-170)*r2*r3
+(666*w2+68*w13+44*w17-6*w26-18*w34-20*w221+46*w442-380)*r2+(-220*w2+40*w13-80*w17-60*w26
+40*w34+280)*r3+(660*w2-100*w13-580*w17-60*w26+460*w34-60*w221+60*w442-380))*t4+(-36*w2
+26*w13+18*w17-10*w34+2*w221-6*w442+42)*r2*r3+(-74*w2-62*w13-26*w17+54*w26+2*w34-26*w221
-14*w442-286)*r2+(-80*w2+90*w13-10*w17-40*w26-10*w221+130)*r3+470*w2+260*w13+140*w17
-110*w26+170*w34-20*w221+30*w442+1060;
c13:=(-20/13*w26*r3*t1*t2+((-20*w2-30*w13-30*w17-12*w26-16*w34-2*w221-42)*r2*r3+(110*w2
+32*w13+56*w17+14*w26+42*w34+16*w221+10*w442+216)*r2)*t1*t3+((20*w2-102*w13-66*w17-48*w26
-44*w34+2*w221-58)*r2*r3+(640*w2-8*w13+56*w17+16*w26+8*w34+88*w221+40*w442
+1048)*r2)*t3)*I+(40/13*w13-20/13*w26)*r3*t1*t2+((-42*w2-44*w13-32*w17-10*w26-10*w34-8*w221
-2*w442-148)*r2*r3+(154*w2+52*w13+36*w17+66*w26+38*w34-4*w221+14*w442-84)*r2)*t1*t3
+((-6*w2-6*w13-18*w17+2*w26-14*w34-22*w221-6*w442-242)*r2*r3+(164*w2+176*w13+48*w17
+148*w26+4*w34-48*w221+4*w442-528)*r2)*t3;
c14:=(((63*w2+23*w26+19*w34+3*w442)*r2*r3+(64*w2-20*w34+4*w442)*r2+(-20*w17+140)*r3-100*w13
-220*w17-20*w221-700)*t1*t3*t4+((-53/2*w2+w13+3*w17-7/2*w26-1/2*w34-w221-3/2*w442
-11)*r2*r3+(-75*w2+26*w13-2*w17+5*w26-5*w34+2*w221-5*w442+22)*r2+(25*w2+25*w13+5*w17
+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221-10*w442
-10)*t1*t3+((-63*w2+19*w13+17*w17-23*w26-19*w34+3*w221-3*w442+73)*r2*r3+(-64*w2-24*w13
-32*w17+20*w34-4*w442+40)*r2+(55*w2+20*w17-25*w26-5*w34-5*w442-140)*r3-620*w2+100*w13
+220*w17-60*w26-180*w34+20*w221-20*w442+700)*t1*t4+((-53/2*w2-11*w13-3*w17-27/2*w26
-21/2*w34-3*w221-3/2*w442-43)*r2*r3+(-44*w2-14*w13-22*w17-12*w26+4*w34-14*w221-4*w442
-174)*r2+(-10*w2+25*w13-15*w17-10*w34+5*w221+85)*r3+(280*w2-40*w13-40*w17+80*w26+140*w34
+20*w442-80))*t1+((108*w2+28*w26+24*w34+8*w442)*r2*r3+(26*w2-22*w26-26*w34+6*w442)*r2
+(-100*w13-20*w17+20*w221+340)*r3-300*w13-540*w17+60*w221-820)*t3*t4+((-46*w2+25*w13
+15*w17+24*w26+12*w34-3*w221-6*w442-13)*r2*r3+(-394*w2+66*w13-2*w17+46*w26-2*w34-18*w221
-34*w442-238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((-108*w2+20*w17-28*w26-24*w34+12*w221-8*w442
+72)*r2*r3+(-26*w2-108*w13-4*w17+22*w26+26*w34+28*w221-6*w442+148)*r2+(-130*w2+100*w13
+20*w17-90*w26+10*w34-20*w221+10*w442-340)*r3-1060*w2+300*w13+540*w17-260*w26-140*w34
-60*w221+20*w442+820)*t4+((-72*w2+20*w13+20*w17-28*w26-14*w34-8*w221-2*w442-128)*r2*r3
+(48*w2+6*w13+18*w17-32*w26+24*w34-46*w221+8*w442-666)*r2+(-140*w2-10*w13-30*w17-20*w26
+40*w34+10*w221+470)*r3+(190*w2-40*w13-400*w17+50*w26+290*w34+40*w221+30*w442+240)))*I
+((25*w13+15*w17+5*w221+75)*r2*r3+(-28*w13-4*w17+4*w221+44)*r2+(35*w2-5*w26-5*w34
-5*w442)*r3+(620*w2+60*w26+180*w34+20*w442))*t1*t3*t4+((95/2*w2+11*w13+13*w17+37/2*w26
+31/2*w34+w221+5/2*w442+31)*r2*r3+(-33*w2-40*w13-40*w17-21*w26-23*w34-12*w221-3*w442
-132)*r2+(55*w2-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17
-80*w26-140*w34-10*w221-20*w442-10)*t1*t3+((43*w2-25*w13-15*w17+11*w26+3*w34-5*w221
+3*w442-75)*r2*r3+(174*w2+28*w13+4*w17+14*w26+22*w34-4*w221+14*w442-44)*r2+(-35*w2-20*w17
99
APPENDIX A.
+5*w26+5*w34+5*w442-20)*r3-620*w2+160*w13+280*w17-60*w26-180*w34+40*w221-20*w442
+560)*t1*t4+((73/2*w2-14*w13-12*w17+19/2*w26+17/2*w34-2*w221+3/2*w442-32)*r2*r3+(20*w2
+42*w13+26*w17-12*w26-16*w34+10*w221+130)*r2+(45*w2+25*w13+5*w17-5*w26+5*w34+5*w221
-5*w442-55)*r3+(340*w2+60*w13+180*w17-20*w26+40*w34+20*w221+620))*t1+((6*w13-2*w17
+10*w221+170)*r2*r3+(-68*w13-44*w17+20*w221+380)*r2+(-220*w2-60*w26+40*w34)*r3+(660*w2
-60*w26+460*w34+60*w442))*t3*t4+((75*w2+23*w13+29*w17+27*w26+21*w34+3*w221+5*w442
-7)*r2*r3+(-130*w2-82*w13-6*w17-42*w26-26*w34-22*w221-10*w442-402)*r2+(-35*w2-90*w13
-10*w17-55*w26-15*w34+10*w221+5*w442+90)*r3-390*w2-260*w13-100*w17-210*w26-130*w34
+60*w221+10*w442+60)*t3+((128*w2-6*w13+2*w17-20*w26-20*w34-10*w221+8*w442-170)*r2*r3
+(666*w2+68*w13+44*w17-6*w26-18*w34-20*w221+46*w442-380)*r2+(220*w2-40*w13+80*w17+60*w26
-40*w34-280)*r3-660*w2+100*w13+580*w17+60*w26-460*w34+60*w221-60*w442+380)*t4+(36*w2
-26*w13-18*w17+10*w34-2*w221+6*w442-42)*r2*r3+(74*w2+62*w13+26*w17-54*w26-2*w34+26*w221
+14*w442+286)*r2+(-80*w2+90*w13-10*w17-40*w26-10*w221+130)*r3+470*w2+260*w13+140*w17
-110*w26+170*w34-20*w221+30*w442+1060;
c15:=(((11/26*w13+1/2*w17+3/26*w221+1/2)*r2*r3+(-10*w2-20/13*w26-10/13*w442)*r2
-40/13*w13*r3)*t1*t2+((43*w2+27*w13+21*w17+11*w26+3*w34+3*w221+3*w442+53)*r2*r3+(174*w2
+24*w13-8*w17+14*w26+22*w34+8*w221+14*w442+88)*r2+(-55*w2-10*w13+10*w17+25*w26+5*w34
-10*w221+5*w442+90)*r3+(620*w2-40*w13+80*w17+60*w26+180*w34+20*w442+680))*t1+((128*w2
+56*w13+28*w17-20*w26-20*w34+4*w221+8*w442+144)*r2*r3+(666*w2+64*w13-48*w17-6*w26-18*w34
-16*w221+46*w442-96)*r2+(130*w2-80*w13+90*w26-10*w34-10*w442-160)*r3+(1060*w2-220*w13
+340*w17+260*w26+140*w34+60*w221-20*w442+940)))*I+((-11/26*w13-1/2*w17-3/26*w221
-1/2)*r2*r3+(-10*w2-20/13*w26-10/13*w442)*r2)*t1*t2+((32*w2-19*w13-17*w17+14*w26+12*w34
-3*w221+2*w442-73)*r2*r3+(-130*w2+24*w13+32*w17-42*w26-26*w34-10*w442-40)*r2+(-85*w2
+20*w17-25*w26+15*w34-5*w442+20)*r3+(80*w2-160*w13-280*w17+40*w26+40*w34-40*w221-560))*t1
+(42*w2-20*w17+26*w26+18*w34-12*w221+2*w442-72)*r2*r3+(-286*w2+108*w13+4*w17-62*w26
-26*w34-28*w221-26*w442-148)*r2+(-470*w2+40*w13-80*w17+10*w26+30*w34-10*w442+280)*r3
-240*w2-100*w13-580*w17+40*w26+400*w34-60*w221-40*w442-380;
c16:=(((-10*w2+4*w26+2*w34)*r2*r3+(24*w2+24*w26+12*w34+4*w442)*r2+(-50*w13+10*w17-10*w221
-30)*r3-20*w13-140*w17-20*w221-540)*t1*t3*t4+((53/2*w2-w13-3*w17+7/2*w26+1/2*w34+w221
+3/2*w442+11)*r2*r3+(75*w2-26*w13+2*w17-5*w26+5*w34-2*w221+5*w442-22)*r2+(25*w2+25*w13
+5*w17+15*w26-5*w34+5*w221+5*w442+25)*r3-350*w2-70*w13-70*w17-50*w26-110*w34-10*w221
-10*w442-10)*t1*t3+((10*w2+27*w13+21*w17-4*w26-2*w34+3*w221+53)*r2*r3+(-24*w2+24*w13
-8*w17-24*w26-12*w34+8*w221-4*w442+88)*r2+(-85*w2+50*w13-10*w17-25*w26+15*w34+10*w221
-5*w442+30)*r3+(80*w2+20*w13+140*w17+40*w26+40*w34+20*w221+540))*t1*t4+((-53/2*w2-11*w13
-3*w17-27/2*w26-21/2*w34-3*w221-3/2*w442-43)*r2*r3+(-44*w2-14*w13-22*w17-12*w26+4*w34
-14*w221-4*w442-174)*r2+(10*w2-25*w13+15*w17+10*w34-5*w221-85)*r3-280*w2+40*w13+40*w17
-80*w26-140*w34-20*w442+80)*t1+((36*w2+28*w26+4*w34-4*w442)*r2*r3+(-122*w2+86*w26-22*w34
-22*w442)*r2+(-80*w13+40*w17-600)*r3-220*w13+260*w17-20*w221-1300)*t3*t4+((46*w2-25*w13
-15*w17-24*w26-12*w34+3*w221+6*w442+13)*r2*r3+(394*w2-66*w13+2*w17-46*w26+2*w34+18*w221
+34*w442+238)*r2+(345*w2+60*w13-60*w17+25*w26-35*w34+5*w442+440)*r3-210*w2+60*w13-420*w17
+10*w26-430*w34-20*w221-10*w442+460)*t3+((-36*w2+56*w13+28*w17-28*w26-4*w34+4*w221+4*w442
+144)*r2*r3+(122*w2+64*w13-48*w17-86*w26+22*w34-16*w221+22*w442-96)*r2+(-470*w2+80*w13
-40*w17+10*w26+30*w34-10*w442+600)*r3-240*w2+220*w13-260*w17+40*w26+400*w34+20*w221
-40*w442+1300)*t4+((-72*w2+20*w13+20*w17-28*w26-14*w34-8*w221-2*w442-128)*r2*r3+(48*w2
+6*w13+18*w17-32*w26+24*w34-46*w221+8*w442-666)*r2+(140*w2+10*w13+30*w17+20*w26-40*w34
-10*w221-470)*r3-190*w2+40*w13+400*w17-50*w26-290*w34-40*w221-30*w442-240))*I+((3*w13
+9*w17-w221-11)*r2*r3+(-56*w13-48*w17-24*w221-304)*r2+(-55*w2+5*w26-15*w34+5*w442)*r3
100
APPENDIX A.
-60*w2+100*w26+100*w34+20*w442)*t1*t3*t4+((-95/2*w2-11*w13-13*w17-37/2*w26-31/2*w34-w221
-5/2*w442-31)*r2*r3+(33*w2+40*w13+40*w17+21*w26+23*w34+12*w221+3*w442+132)*r2+(55*w2
-5*w13-5*w17-15*w26+5*w34-5*w221-5*w442+115)*r3-280*w2-70*w13-70*w17-80*w26-140*w34
-10*w221-20*w442-10)*t1*t3+((32*w2-3*w13-9*w17+14*w26+12*w34+w221+2*w442+11)*r2*r3
+(-130*w2+56*w13+48*w17-42*w26-26*w34+24*w221-10*w442+304)*r2+(55*w2+10*w13-10*w17-5*w26
+15*w34+10*w221-5*w442-90)*r3+(60*w2+40*w13-80*w17-100*w26-100*w34-20*w442-680))*t1*t4
+((73/2*w2-14*w13-12*w17+19/2*w26+17/2*w34-2*w221+3/2*w442-32)*r2*r3+(20*w2+42*w13+26*w17
-12*w26-16*w34+10*w221+130)*r2+(-45*w2-25*w13-5*w17+5*w26-5*w34-5*w221+5*w442+55)*r3
-340*w2-60*w13-180*w17+20*w26-40*w34-20*w221-620)*t1+((46*w13+38*w17-6*w221-86)*r2*r3
+(-56*w13-8*w17-72*w221-952)*r2+(-60*w2+20*w26+40*w34)*r3-280*w2+160*w26+120*w34)*t3*t4
+((-75*w2-23*w13-29*w17-27*w26-21*w34-3*w221-5*w442+7)*r2*r3+(130*w2+82*w13+6*w17+42*w26
+26*w34+22*w221+10*w442+402)*r2+(-35*w2-90*w13-10*w17-55*w26-15*w34+10*w221+5*w442+90)*r3
-390*w2-260*w13-100*w17-210*w26-130*w34+60*w221+10*w442+60)*t3+((42*w2-46*w13-38*w17
+26*w26+18*w34+6*w221+2*w442+86)*r2*r3+(-286*w2+56*w13+8*w17-62*w26-26*w34+72*w221
-26*w442+952)*r2+(60*w2+80*w13-20*w26-40*w34+160)*r3+(280*w2+220*w13-340*w17-160*w26
-120*w34-60*w221-940))*t4+(36*w2-26*w13-18*w17+10*w34-2*w221+6*w442-42)*r2*r3+(74*w2
+62*w13+26*w17-54*w26-2*w34+26*w221+14*w442+286)*r2+(80*w2-90*w13+10*w17+40*w26+10*w221
-130)*r3-470*w2-260*w13-140*w17+110*w26-170*w34+20*w221-30*w442-1060;
phi:=[ c01, c02, c03, c04, c05, c06, c07, c08, c09, c10, c11, c12, c13, c14, c15, c16];
101
Index
Hd , 31
LR
d (π, v), 53
Zu , 37
inner product, 9
absolute bound, 12
almost flat, 53
automorphism
quantum design, 30
vectors, 41
monomial, see also phase-permutation, 51
character, 17
non-principal, 17
principal, 17
Clifford group, 46
complete set of MUBs, 22
difference set, 15
planar, 26
Singer, 26
equiangular, 10
exact solutions, 2
fiducial vector, 3, 38
flat, 26
generalized Pauli group, 36
group algebra, 15
group covariant, 4
design, 32
vectors, 32
Hadamard matrix
complex, 18, 63
real, 18
Hermitian, 10
left invariant, see automorphism vectors
norm, 9
numerical solutions, 2
orthogonal, 9
basis, 18
projection, 10
outer product, 9
Pauli group, 28
Pauli matrices, 28
phase-permutation matrices, see also monomial, 49
projection matrix, 10
quantum design, 29
degree, 29
maximal, 29
regular, 29
RDS, 16
regular
automorphism subgroup, 30
quantum design, 29
relative difference set, 16
restriction, 17
SIC-POVM, 2, 43
tight frame, 45
unbiased bases, 19
MUBs, 19
102
INDEX
unitary, 30
Weyl matrices, 31
Weyl-Heisenberg group, 31
Zauner’s unitary, 37
103