Spec. Matrices 2014; 2:68–77
Special Matrices
Research Article
Open Access
Rajesh Pereira* and Joanna Boneng
The theory and applications of complex
matrix scalings
Abstract: We generalize the theory of positive diagonal scalings of real positive definite matrices to complex
diagonal scalings of complex positive definite matrices. A matrix A is a diagonal scaling of a positive definite
matrix M if there exists an invertible complex diagonal matrix D such that A = D* MD and where every row
and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and
we conjecture that every n by n positive definite matrix has at most 2n−1 scalings and prove this conjecture for
certain special classes of matrices. We also use the theory of complex diagonal matrix scalings to formulate a
van der Waerden type question on the permanent function; we show that the solution of this question would
have applications to finding certain maximally entangled quantum states.
Keywords: Diagonal Matrix Scalings, Positive Definite Matrices, Circulant Matrices, Tridiagonal Matrices,
Doubly Stochastic Matrices, Permanent; Symmetric States, Geometric Measure of Entanglement
MSC: 15B48, 15B51, 81P40
DOI 10.2478/spma-2014-0007
Received December 5, 2013; accepted April 13, 2014.
1 Introduction
In this paper we develop the complex version of the theory of diagonal matrix scalings of positive definite
matrices first explored by Marshall and Olkin [10]. In this section, we review the properties of some of the
classes of matrices we will use in the paper. In Section two, we briefly describe the relevant past results on
diagonal matrix scalings and explore the theory of complex diagonal matrix scalings. In particular, we formulate a conjecture on the number of different matrix scalings a positive definite matrix could have; in Section
three we prove this conjecture for certain special classes of matrices. In Section four, we use the theory of
matrix scalings to state a van der Waerden-type question about permanents and show that this question has
applications to the study of highly entangled quantum states.
We begin by reviewing some of the matrix theoretical definitions used in this paper.
Definition 1.1. An n by n complex matrix A is said to be positive definite if A = A* and x* Ax > 0 for all non-zero
x ∈ Cn . An n by n complex matrix A is said to be positive definite if A = A* and x* Ax ≥ 0 for all x ∈ Cn .
Definition 1.2. Let n ∈ N and let ω = e
entry is √1n ω(j−1)(k−1) .
2πi
n
. Then the nth Fourier matrix F n is the n by n matrix whose (j, k)th
It can be easily verified that F n is a unitary matrix for all n.
*Corresponding Author: Rajesh Pereira: Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada
N1G 2W1, E-mail: [email protected]
Joanna Boneng: Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1
© 2014 Rajesh Pereira and Joanna Boneng, licensee De Gruyter Open. This work is licensed under the Creative Commons AttributionNonCommercial-NoDerivs 3.0 License.
Unauthenticated
Download Date | 6/18/17 11:13 PM
The theory and applications of complex matrix scalings |
69
Definition 1.3. Let M be an n by n complex matrix. We say that M is an equisum matrix if there exists an r ∈ R
such every row and every column of M consists of elements which sum to r. If further r = 1, we say that M is
a doubly quasi-stochastic matrix. A doubly quasi-stochastic matrix all of whose entries are nonnegative real
numbers is called a doubly stochastic matrix.
The set of all n by n complex positive definite doubly quasi-stochastic matrices will play a key role in this
paper; this set will be denoted QSP n . The term equisum is from [3]. We will use the following observation
about equisum matrices from the same paper.
Proposition 1.1. [3] Let M be an n by n matrix and let A = F *n MF n . Then M is an equisum matrix if and only if
every entry in both the first row and the first column of A, except possibly a11 , is zero. In this case, a11 will be
the sum of each row and each column of M.
We now introduce a particularly useful class of equisum matrices.
Definition 1.4. An n by n matrix A is called a circulant matrix if there exist n complex numbers c0 , c1 , ..., c n−1
such that a ij = c i−j where the subtraction in the subscript is taken mod n.
Much more about circulant matrices can be found in reference [2], but we will only use the fact that the
circulants are diagonalized by the Fourier matrix of the appropriate size. Circulant matrices are a special
type of Toeplitz matrix; the definition of Toeplitz matrices follows:
Definition 1.5. An n by n matrix A is called a Toeplitz matrix if there exist n complex numbers
c−(n−1) , c−(n−2) , ..., c−1 , c0 , c1 , ..., c n−1 such that a ij = c i−j .
The final class of matrices we review are the tridiagonal matrices. A matrix A is tridiagonal if a ij = 0 whenever
|i − j| ≥ 2. Neither Toeplitz nor tridiagonal matrices are necessarily equisum.
2 Complex Diagonal Scalings
The theory of matrix scalings was founded by Sinkhorn in [15] where it was proved that if A is any n by n matrix
with strictly positive entries, then there exists two positive diagonal matrices D1 and D2 such that D1 AD2 is
a doubly stochastic matrix. Much work was done to extend Sinkhorn’s result to other classes of matrices; a
good summary of the work done in this area can be found in the first section of [8]. One notable result was
that of Marshall and Olkin [10], who gave a variant of Sinkhorn’s result for real positive definite matrices.
More specifically, Marshall and Olkin showed that if A is an n by n real positive definite matrix, then there
exists a unique n by n positive diagonal matrix D such that DAD is a doubly stochastic matrix. Motivated by
problems involving the zeros of polynomials, Marshall and Olkin’s result was extended to complex positive
definite matrices in [12, Lemma 2.9] where it was shown that if A is an n by n complex positive definite matrix,
then there exists an n by n complex invertible diagonal matrix D such that D* AD is a doubly quasi-stochastic
matrix. This motivates the following definitions:
Definition 2.1. Let M be an n by n positive definite complex matrix. We say that a matrix A is a diagonal
scaling of M if A is a doubly quasi-stochastic matrix and there exists an n by n complex invertible diagonal
matrix D such that A = D* MD. Let sc(M) be the cardinality of the set of diagonal scalings of M. (In other words,
sc(M) = #{D* MD ∈ QSP n : D is a complex invertible diagonal matrix } ).
We note that for any given positive definite matrix M, there may be many invertible diagonal matrices which
yield the same scaling. Suppose there exist two diagonal matrices D and E such that D* MD = E* ME. We will
show that if M has no zero entries, then these diagonal matrices must be scalar multiples of one another.
Unauthenticated
Download Date | 6/18/17 11:13 PM
70 | Rajesh Pereira and Joanna Boneng
Proposition 2.1. Let M ∈ M N (C) have no zero entries. Suppose there exist two invertible diagonal matrices D
and E with D* MD = E* ME, then there exists a ω ∈ C with |ω| = 1 such that E = ωD.
Proof. Let d1 , ..., d n and e1 , ..., e n be the main diagonal entries of D and E respectively. Then d i m ij d j =
e i m ij e j for all i, j. Let c k = de kk for all k, then c i = c1j for all 1 ≤ i, j ≤ n. Setting i = j, we get |c i | = 1 for
1 ≤ i ≤ n. Hence c j = (c i )−1 = c i for all 1 ≤ i, j ≤ n.
Hence for M positive definite with no zero entries, sc(M) is also equal to number of invertible diagonal matrices D unique up to scalar multiplication such that D* MD ∈ QSP n . This result fails for matrices M with zero
entries, which is why we focus on the number of diagonal scalings rather than on the number of diagonal
matrices which implement those scalings.
In this paper we explore the set of complex diagonal scalings of complex positive definite matrices. We
start by restating one component of the proof of [12, Lemma 2.9] in a convenient equivalent form. We use say
that a function has a local extremum at a point if it has either a local maximum or a local minimum at that
point.
Lemma 2.1. Let A be an n by n complex positive definite matrix and e be the n-vector all of whose entries are
one. Let ∆ n be the set of all invertible complex diagonal matrices and let f : ∆ n → R be defined as f (D) =
e T D* ADe
*
2 . If D is a local extremum of f , then D AD is a positive definite equisum matrix.
|det(D)| n
We note that the proof of this result is nearly identical to part of the proof of [12, Lemma 2.9]. We provide a
proof here for the convenience of the reader.
Proof. Suppose D = diag(d1 , d2 , ..., d n ) is a local extermum of f . Let S = {D ∈ ∆ n : |det(D)| = 1}. Since
f (cD) = f (D) for any c > 0, we may assume D ∈ S. For 1 ≤ k ≤ n, let r k be the sum of the elements in the kth
row of D* AD and let s k = r k − |d k |2 a kk . Let D(t) = diag(e it d1 , d2 , ..., d n ) for all t ∈ R. Then e T D(t)* AD(t)e =
f (D) + 2Re([e it − 1]s1 ) and since this function has a local extremum at t = 0, s1 and hence r1 must be real.
A similar argument shows us that r k is real for all k and hence r k is also the sum of the elements in the kth
column of D* AD. Let D(ϵ) = diag((1 − ϵ)d1 , (1 − ϵ)−1 d2 , d3 , d4 , ..., d n−1 , d n ). Expanding (1 − ϵ)−1 as a power
series we get e T D(ϵ)* AD(ϵ)e = f (D) + 2(r2 − r1 )ϵ + O(ϵ2 ). Since this function has a local extremum at ϵ = 0,
r2 = r1 . Hence all row and column sums are equal and D* AD is an equisum matrix.
We can then use the fact that both the numerator and denominator of f are homogeneous functions to give a
further equivalent statement of this result which will prove useful in a later section.
Corollary 2.1. Let A be an n by n complex positive definite matrix and let S be the ellipsoid E = {v ∈ Cn :
Q
v* Av = n}. Let g : E → R be defined as g(z1 , z2 , ..., z n ) = | nk=1 z k |. If v ∈ E is a local maximum of g in E and
D = diag(v1 , v2 , ..., v n ), then D* AD is a doubly quasi-stochastic matrix.
Proof. Let v = (v1 , v2 , ..., v n ) ∈ E be a local maximum of g in E. Let D = diag(v1 , v2 , ..., v n ). Then it is clear
that D is a local minimum for the f in the previous lemma, so D* AD is equisum. Since e T D* ADe = v* Av = n,
D* AD is doubly quasi-stochastic.
We note that since E is a compact set and g is continuous, the existence of a local maximum is guaranteed which means that every complex positive definite matrix A can be diagonally scaled to a doubly quasistochastic matrix. For any fixed A, there may be several complex diagonal matrices D for which D* AD is a
doubly quasi-stochastic matrix and different choices for D may give us a different doubly quasi-stochastic
matrix D* AD. Hence we have sc(M) ≥ 1 for any positive definite matrix M. It is clear that sc(M) = 1 whenever
M is a diagonal matrix, since the identity is the only quasi-stochastic diagonal matrix. We state the following
conjecture motivated by some partial results given in the next section.
Conjecture 2.1. Let M be an n by n complex positive definite matrix, then sc(M) ≤ 2n−1 .
Unauthenticated
Download Date | 6/18/17 11:13 PM
The theory and applications of complex matrix scalings |
71
There are other interesting questions that can be raised about the geometry of the set of scalings of a given
complex positive matrix. In this paper, we largely focus on the number of distinct scalings as the first step
toward the study of the geometry of the set of scalings. In the next section, we will prove this conjecture for all
two by two matrices, three by three circulant matrices and real tridiagonal matrices of arbitrary size. For the
remainder of this section, we continue exploring properties of matrix scalings. We begin with a well known
elementary result from positive scalings of real matrices and note that it also holds for complex scalings of
complex matrices with an identical proof.
Proposition 2.2. [10] Let A be a complex positive definite matrix, let D = diag(d1 , d2 , ..., d n ) be a complex
invertible diagonal matrix and let v = (d1 , d2 , ...d n )T be the vector of main diagonal entries of D. Then D* AD
is a diagonal scaling of A if and only if Av = w where w = (1/d1 , 1/d2 , ..., 1/d n )T .
If we assume |d1 | = |d2 | = ... = |d n | = c, we get d i = c2 /d i for all i ∈ {1, ..., n}. Thus v = c2 w. Hence if D is
a unitary diagonal matrix, then D* AD is a diagonal scaling of positive definite matrix A if and only if v is an
eigenvector of A. Since a circulant matrix has an eigenbasis consisting entirely of vectors having all entries of
modulus √1n ; this provides us with up to n different scalings of an n by n circulant matrix; these scalings are
also circulants as well. If Conjecture 2.1 holds, we might have up to 2n−1 − n other scalings when n ≥ 3, these
scalings may not necessarily be circulants.
Since we know that every positive definite matrix has at least one scaling, we can assume the matrix is already scaled to be doubly quasi-stochastic (and hence in QSP n ) and look at rescalings. One may ask which diagonal matrices D have the property that they rescale some matrix in QSP n . Using Proposition 2.2, we see that
for any D = diag(d1 , d2 , ..., d n ) there exists an A ∈ QSP n such that D* AD ∈ QSP n if and only if there is a n by
n positive definite matrix which maps (1, 1, ..., 1)T to itself and (d1 , d2 , ..., d n )T to (1/d1 , 1/d2 , ..., 1/d n )T .
This is a special case of the positive definite interpolation problem: given two sets of linearly independent
vectors {v i }ki=1 and {w i }ki=1 in Cn , does there exist an n by n positive definite matrix A such that Av i = w i for
all i? This problem has been solved by Pinkus whose solution we give below.
Theorem 2.1. [13, Theorem 2.1] Let {v i }ki=1 and {w i }ki=1 be two sets of linearly independent vectors in Cn . Then
there exists an n by n Hermitian positive definite matrix A satisfying Av i = w i ∀i ∈ {1, ..., k} if and only if the
P
P
inner product h c i v i , c j w j i > 0 for all set of complex numbers {c i }ki=1 which are not all zero.
Applying this result to our specific case we obtain the following:
some positive definite doubly

 D rescales
n
P
n
dj


i=1
 is positive definite. We can

quasi-stochastic matrix if and only if the two by two matrix P
n

1
n
dj
j=1
rephrase this condition as follows:
Proposition 2.3. Let D = diag(d1 , d2 , ..., d n ) be a non-scalar diagonal matrix. Then there exists a matrix
n
n
n
n
P
P
P
1
1 P
A ∈ QSP n such that D* AD ∈ QSP n if and only if both d j =
d j ) < n2 .
d j and (
d j )(
i=1
j=1
j=1
j=1
As a corollary, we can give an easy proof that any complex positive definite matrix has either zero or one
positive scalings. This is known in the real case [10] and is also a simple consequence of [8, Theorem 2]. We
first need the Chebyshev sum inequality for monotone sequences. (See [6] and the references therein for more
details on this inequality).
Lemma 2.2. (Chebyshev sum inequality) Let {a k }nk=1 and {b k }nk=1 be two n-tuples of positive numbers in inP
P
P
creasing and decreasing order respectively, then n nk=1 a k b k ≤ ( nk=1 a k )( nk=1 b k ).
Proposition 2.4. Let A be an n by n complex positive definite matrix, then there exists at most one positive
diagonal matrix D such that DAD ∈ QSP n .
Unauthenticated
Download Date | 6/18/17 11:13 PM
72 | Rajesh Pereira and Joanna Boneng
Proof. Suppose there exists two different positive diagonal matrices D1 and D2 such that both D1 AD1 and
D2 AD2 are doubly quasi-stochastic. Let B = D1 AD1 and D = D−1
1 D 2 , then D 2 AD 2 = DBD. Let d 1 , d 2 , ..., d n be
n
n
P
P
the main diagonal entries of D reordered so that they are in increasing order. We must have ( d1j )( d j ) < n2 ,
j=1
2
but Chebyshev’s inequality gives us n ≤
n
P
n
P
( d1j )( d j )
j=1
j=1
j=1
which is a contradiction.
3 Evidence for the main conjecture
In this section we prove certain special cases of Conjecture 2.1. We begin by defining a natural equivalence
relation on QSP n .
Definition 3.1. We say that A, B ∈ QSP n are equivalent if there exists an n by n invertible complex diagonal
matrix D such that B = D* AD.
We can rephrase our main conjecture in terms of equivalence classes in QSP n as follows:
Conjecture 3.1. Every equivalence class in QSP n has at most 2n−1 members.
We now prove our conjecture for certain special classes of matrices. We begin with
! the easiest case. Every two
a b
by two equisum matrix is a circulant matrix of the form circ(a, b) =
. This matrix will be positive
b a
definite if and only if a and b are both real a > |b| > 0, and will be doubly quasi-stochastic if a + b = 1. In this
b
a
, − a−b
)} and every equivalence class in
case, the equivalence class of circ(a, b) is clearly {circ(a, b), circ( a−b
QSP2 consists either of two circulants or of just the identity matrix.
In higher dimensions, not every equisum matrix is a circulant. While we do not have a general description
of equivalence classes in QSP n , we can show that every equivalence class containing a non-scalar circulant
has three or four elements, three of which are circulants thereby proving Conjecture 2.1 in this case.
Theorem 3.1. Let M be a three by three positive definite circulant matrix, then sc(M) ≤ 4.
Proof. Let M be a three by three positive definite circulant matrix, D be a three by three diagonal matrix and
F be the three by three Fourier matrix. Then F * MF is a diagonal matrix and F * DF is a circulant matrix. Let
B = F * MF and C = F * DF. Then C* BC = (F * D* F)(F * MF)(F * DF) = F * D* MDF. If D* MD is a doubly quasistochastic matrix, Proposition 1.1 shows us that C* BC = F * (D* MD)F has every entry equal to zero in the first
row and column except for the (1, 1) entry which will be one. We can restate our scaling problem for the three
by three circulants as follows: given a three by three positive definite diagonal matrix D find all circulant
matrices C such that C* DC has every entry equal to zero in the first row and column except for the one in the
(1, 1) entry. 



x y z
a 0 0




Let C =  z x y  and B = 0 b 0. Note that a, b, c are the eigenvalues of M. We now find the
y z x
0 0 c
first row of C* BC in terms of x, y, z, a, b, c. The first column is the transpose of the following vector:


a|x|2 + b|z|2 + c|y|2


 axy + bxz + cyz 
axz + byz + cxy
Then we get the system
0 = axy + bxz + cyz
(1)
Unauthenticated
Download Date | 6/18/17 11:13 PM
The theory and applications of complex matrix scalings |
73
0 = axz + byz + cxy
(2)
0 = cxy + axz + byz
(3)
Take the complex conjugate of (2)
Then the system of equations (1) and (3) can be written in matrix form:
a
c
b
a
c
b
!


xy
 
 xz  =
yz
0
0
!
Hence the vector (xy, xz, yz)T is in the null space of a real matrix and so can be taken to be a real vector.
Hence x, y, z can be taken to be real. The vector v = (xy, xz, yz)T is orthogonal to both (a, b, c) and (c, a, b).
Hence v must be a multiple of (b2 − ac, c2 − ab, a2 − cb) which is the cross product of these two vectors. One
choice is v = 0 which will occur when any two of x, y, z are zero. These give us the scalings D* MD where
either D = diag(1, 1, 1), D = diag(1, ω, ω2 ) or D = diag(1, ω2 , ω). We can also get these scalings from the
eigenvectors of circulants by the remark following Proposition 2.2. We now consider the case where v ≠ 0,
xy = k(b2 − ac)
(4)
xz = k(c2 − ab)
(5)
yz = k(a2 − bc)
(6)
If any one of the right hand sides of (4), (5) and (6) is zero, then we get no further solutions so suppose
none of these are zero. We note that the magnitude of k is fixed by the condition that a|x|2 +b|z|2 +c|y|2 = 1 and
we know show that the sign of k is fixed as well. We note that at least one of these terms on the right hand side
will be positive and at least one of the terms will be negative (since if a ≥ b ≥ c > 0, then (a2 −bc) is positive and
(c2 − ab) is negative). Since the product of these terms is equal to (xyz)2 , one of these factors must be positive
and two must be negative, this fixes the sign of k and hence fixes the exact value of k. Taking the absolute
values and then the logarithms of these equations yield the following linear equations in ln(x), ln(y), ln(z).
ln(|x|) + ln(|y|) = ln(|k||b2 − ac|)
(7)
ln(|x|) + ln(|z|) = ln(|k||c2 − ab|)
(8)
ln(|y|) + ln(|z|) = ln(|k||a2 − bc|)
(9)
We can now use standard algebra to obtain a unique solution for |x|, |y|, |z|.
r
k(b2 − ac)(c2 − ab)
|
|x| = |
a2 − bc
r
k(b2 − ac)(a2 − bc)
|y| = |
|
c2 − ab
r
k(a2 − bc)(c2 − ab)
|z| = |
|
b2 − ac
If we fix the sign of x, the signs of y and z are determined by the signs of the right hand sides of (4),
(5) and (6). Hence this solution gives rise to a unique scaling. So sc(M) ≤ 4 whenever M is a three by three
circulant.
Unauthenticated
Download Date | 6/18/17 11:13 PM
74 | Rajesh Pereira and Joanna Boneng
We note that our proof means that we have an exact value for sc(M) when M is a three by three positive
definite circulant. If M is a multiple of the identity then sc(M) = 1, if M is not a multiple of the identity, then
1
1
sc(M) = 3 if det(M) 3 is an eigenvalue of M and sc(M) = 4 if det(M) 3 is not an eigenvalue of M.
We note that every element of QSP2 is a real matrix; this is not true of QSP3 where even most real three
by three circulants share their equivalence class with nonreal matrices. What distinguishes the two by two
case from higher dimensions is that every 2 by 2 matrix is a real tridiagonal matrix. We will see that all real
tridiagonal matrices have the same property; if T is an n by n real tridiagonal matrix, then any D* TD ∈ QSP n
must also be a real matrix.
Proposition 3.1. Any equivalence class in QSP n which contains a real tridiagonal matrix has at most 2n−1
members and consists entirely of real tridiagonal matrices.
Proof. Let T be a real tridiagonal matrix. Then an invertible diagonal matrix D scales T if and only if Tv = w
where v = (d1 , d2 , ..., d n ) and w = (1/d1 , 1/d2 , ..., 1/d n )T . By multiplying D by a complex scalar of modulus
one, we may assume d1 is real. Using the first row of the equation Tv = w, we can solve for d2 as a real linear
combination of d1 and d−1
1 , hence d 2 is real. Using the second row of the equation Tv = w, we can solve for d 3
as a real linear combination of d1 ,d2 and d−1
2 . Continuing down the rows of Tu = w, we can show that all of
the elements of D are real by induction. Now suppose T is in QSP n as well as being real tridiagonal. Suppose
D1 and D2 are two diagonal matrices which scale T and have the same sign patterns then D−1
1 D 2 is the unique
positive diagonal matrix which scales T and hence D−1
1 D 2 must be the identity by Proposition 2.4. A similar
argument shows that if D1 and D2 have the opposite sign in each diagonal entry, we must have D1 = −D2 and
hence D1 TD1 = D2 TD2 . Therefore the equivalence class containing T has at most 2n−1 members.
This proves our conjecture for the special case of the real tridiagonal matrices.
4 Permanents and the geometric measure of entanglement
In this section we explore connections between the permanent function and the diagonal scalings of positive
definite matrices. We will later look at an application to the study of the geometric measure of entanglement
of symmetric states. Let S n denote the symmetric group of order n. Recall that for any A ∈ M n (C), the perP
Q
manent of A is defined as per(A) = σ∈S n nk=1 a kσ(k) [11]. One of the most well-known conjectures in matrix
theory was the van der Waerden conjecture about the lower bound of the permanent function on the doubly
stochastic matrices which was proved independently in the early eighties by Egorychev [4] and Falikman [5].
We state their result together with the much more elementary upper bound.
Theorem 4.1. [4, 5] Let S be an n by n doubly stochastic matrix, then per(S) ∈ [ nn!n , 1]. The lower bound is
attained only when S is the matrix all of whose entries are 1n and the upper bound is attained when S is a permutation matrix.
We can consider the same problem for positive semidefinite doubly quasi-stochastic matrices in M n (C). In
this case we get the same lower bound (this was proved even earlier than the van der Waerden conjecture
!
1+x
−x
by Marvin Marcus [9, theorem 2]) but no upper bound at all. To see this consider the matrix
−x
1+x
which is positive definite doubly quasi-stochastic for all x ≥ − 12 and has permanent 1 + 2x + 2x2 which goes
to infinity as x gets large.
We note however that the upper bound becomes interesting when we consider equivalence classes in
QSP n . We introduce the following definition:
Unauthenticated
Download Date | 6/18/17 11:13 PM
The theory and applications of complex matrix scalings |
75
Definition 4.1. Let M ∈ QSP n . We call M minimally scaled if whenever D is a complex diagonal matrix such
that D* MD ∈ QSP n we have per(D* MD) ≥ per(M). We call M maximally scaled if whenever D is a complex
diagonal matrix such that D* MD ∈ QSP n we have per(D* MD) ≤ per(M). We let MinSc n denote the set of
minimally scaled matrices in QSP n and we let MaxSc n denote the set of maximally scaled matrices in QSP n .
In other words, a minimally scaled matrix has the smallest permanent in its equivalence class. We note that
the inverse of a matrix in QSP n is also in QSP n . Hence if D* MD is a scaling of M, D−1 M −1 (D* )−1 is a scaling of
Q
M −1 . Since per(D* MD) = per(A) nk=1 |d kk |2 , we can observe the following:
Proposition 4.1. Let M be an positive definite matrix. D* MD is a minimal scaling of M if and only if D−1 M −1 (D* )−1
is a maximal scaling of M −1 .
We can ask the following question about the range of the permanent on MinSc n .
Question 4.1. What is the structure of MinSc n and MaxSc n ? What are the element(s) in MinSc n which have
the largest permanent? Which are the element(s) of MaxSc n which have the smallest permanent?
These questions have a simple answer when n = 2. In this case, every equivalence class has at most two
members, one of which is doubly stochastic. Direct calculation shows us that the doubly stochastic matrix
has the smaller permanent of the two matrices in the equivalence class. Hence MinSc2 is exactly the two by
two positive definite doubly stochastic matrices and the largest permanent in MinSc2 is achieved by the two
by two identity. The same idea shows us that the element in MaxSc2 which has smallest permanent is also
the two by two identity. This idea fails when n ≥ 3 as there are equivalence classes containing no doubly
stochastic matrices. An example is the equivalence class containing the following matrix which cannot be
scaled to a doubly stochastic matrix because of the signs of its entries:

1
 1
− 4
1
4
− 41
1
1
4

1
4
1
4
1
2
We will now present some motivation for Question 4.1 by showing that the elements in MinSc n which
have maximal permanent can be used to generate highly entangled symmetric states. We start by introducing some concepts from quantum physics beginning with the geometric measure of entanglement. We will
concentrate entirely on the mathematical properties of this concept; readers interested in the physical background can consult [1, 7, 16]. Any composite pure state in quantum mechanics can be represented as an unit
length element of the tensor product of Hilbert spaces. Certain composite systems have the property that
measuring one component of the system simultaneously affects the other components of the system. This
property is called entanglement. Entanglement is a key resource in quantum computation. Different systems
have different degrees of entanglement and many different measures of entanglement have been proposed [1,
Ch. 15]. The geometric measure of entanglement was proposed by Abner Shimony [14]; it takes advantage of
the fact that we have a simple mathematical characterization of the states which possess no entanglement.
Definition 4.2. Let {H k }nk=1 be complex Hilbert spaces and let |ϕi ∈
N
if there exists |ϕ k i ∈ H k such that |ϕi = nk=1 |ϕ k i.
Nn
k=1
H k . Then |ϕi is said to be separable
Shimony’s insight was that the distance between a state and the set of separable states could be used as
measure of entanglement. This idea was extended in [16] where Wei and Goldbart used the minimal angle
between the state and the set of separable states to give an equivalent and more common definition of the
geometric measure of entanglement. The idea is that if θ is the smallest angle between |ϕi and the set of
symmetric states, the geometric measure of entanglement is sin2 (θ) = 1 − cos2 (θ). This measure can easily be
expressed as a function of |ϕi.
Unauthenticated
Download Date | 6/18/17 11:13 PM
76 | Rajesh Pereira and Joanna Boneng
Definition 4.3. Let {H k }nk=1 be complex Hilbert spaces and let Sep be the set of unit length separable states
N
N
in nk=1 H k . Let |ϕi be an arbitrary unit length element in nk=1 H k . Then we define the geometric measure of
entanglement of |ϕi as follows: E(|ϕi) = 1 − max|ψi∈Sep |hϕ|ψi|2 .
We will examine a possible approach to finding the geometric measure of a certain class of states called the
symmetric states. We denote by H ⊗n the tensor product of n identical copies of a Hilbert space H and denote
by |ψi⊗n the tensor product of n identical copies of |ψi.
Definition 4.4. Let H be a complex Hilbert space and let |ϕi be a unit length element in H ⊗n . Then |ϕi is said
Nn
1 P
to be a symmetric state if there exists |ϕ1 i, |ϕ2 i, ..., |ϕ n i ∈ H such that |ϕi = n!
σ∈S n
k=1 | ϕ σ(k) i.
We note that any symmetric separable state in H ⊗n must be of the form |ψi⊗n for some unit length |ψi ∈ H.
It was shown in [7] that the closest separable state to a symmetric state must be a symmetric separable state.
This greatly simplifies the calculation of the geometric measure of entanglement for symmetric states.
P
N
Proposition 4.2. Let H be a complex Hilbert space, let |ϕ1 i, |ϕ2 i, ..., |ϕ n i ∈ H and let |ϕi = σ∈S n nk=1 |ϕ σ(k) i
Q
be a unit length symmetric state in H ⊗n . Then 1 − E(|ϕi) = max|ψi nk=1 |hψ|ϕ k i|2 where the maximum is taken
over all unit vectors |ψi in H.
We now introduce our main tool we use to analyze symmetric states.
Nn
1 P
Definition 4.5. Let H be a complex Hilbert space and let |ϕi = n!
σ∈S n
k=1 | ϕ σ (k)i be a symmetric state
⊗n
in H . We define a Gram matrix of |ϕi to be the the n by n matrix G ϕ whose (i, j)th entry is the inner product
h ϕ i | ϕ j i.
Nn
1 P
We note that a Gram matrix of |ϕi depends on the representation |ϕi = n!
σ∈S n
k=1 | ϕ σ(k) i and hence is not
unique; P T D* G ϕ DP will be another Gram matrix of |ϕi for any permutation matrix P and any diagonal matrix
D with determinant one. For our purposes the choice of Gram matrix will not matter and we will choose any
one of these. The Gram matrix is clearly positive semidefinite and the permanent of the Gram matrix will be
n!hϕ|ϕi = n!.
We now restrict ourselves to the case where H = Cn and consider the geometric entanglement of symmetric states in (Cn )⊗n (finite dimensional Hilbert spaces are used in quantum computation, so even this
restricted case is of considerable interest). If we choose B to be the n by n matrix whose kth column vectors
is |ϕ k i for all k, then G ϕ = B* B. Now let |ψi be a unit vector in Cn and v T = (hψ|ϕ1 i, hψ|ϕ2 i, ..., hψ|ϕ n i). If
G ϕ is invertible, then so is B, in this case v T B−1 is the transpose of ψ and hence a unit vector for any choice of
Qn
T −1
T −1 *
2
ψ. Hence we obtain the condition v* G−1
ϕ v = (v B )(v B ) = 1. We note that 1 − E(| ϕ i) = max{ k=1 | v k | :
v* G−1
ϕ v = 1}. We now can use Corollary 2.1 to restate the geometric measure of entanglement as a scaling
problem.
Theorem 4.2. Let |ϕi be a symmetric state in (Cn )⊗n whose Gram matrix G ϕ is invertible. Let D be an n by
n invertible diagonal matrix such that M = D* G ϕ D is a minimally scaled doubly quasi-stochastic matrix. Then
n!
.
E(|ϕi) = 1 − (n n )per(M)
Q
√
ndiag(v1 , v2 , ..., v n ). Then
Proof. Let v ∈ Cn be such that 1 − E(|ϕi) = nk=1 |v k |2 and v* G−1
ϕ v = 1. Let A =
* −1
*
by Corollary 2.1, A* G−1
A
is
maximally
scaled.
Let
D
=
(A
)
and
let
M
=
D
G
D.
By Proposition 4.1, M is
ϕ
ϕ
Q
n!
minimally scaled. Clearly per(M) = n−n nk=1 |v k |−2 per(G ϕ ) = nn!n (1 − E(|ϕi))−1 and E(|ϕi) = 1 − (n n )per(M)
.
We note that the lower bound inequality per(M) ≥ nn!n gives us E(|ϕi) ≥ 0. The n by n minimally scaled matrix
1
n!
M with the largest permanent is of great interest since kM (where k = ( per(M)
) n ) will be the Gram matrix of the
symmetric state in (Cn )⊗n with largest geometric measure of entanglement. This state could then be easily
found from the Gram matrix using the Cholesky decomposition.
Unauthenticated
Download Date | 6/18/17 11:13 PM
The theory and applications of complex matrix scalings |
77
Acknowledgement: The authors would like to thank Prof. Pal Fischer for bringing Lemma 2.2 to their attention. Both authors would like to acknowledge NSERC support in the form of NSERC Discovery Grant 400550
for the first author and a NSERC Undergraduate Student Research Award for the second author. The authors
would like to thank the referees for their careful reading of the paper and for their many suggestions which
have greatly improved the paper.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
I. Bengtsson and K. Zyczkowski. Geometry of Quantum States. Cambridge University Press, 2006.
P. J. Davis. Circulant Matrices. John Wiley & Sons, 1979.
P. J. Davis and I. Najfeld. Equisum matrices and their permanence. Quart. Appl. Math., 58(1):151–169, 2000.
G. P. Egorychev. The solution of van der Waerden’s problem for permanents. Adv. Math., 42:299–305, 1981.
D. I. Falikman. A proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki,
29:931–938, 1981.
G. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cambridge Mathematical Library, 1952.
R. Hubener, M. Kleinmann, T. Wei, C. Gonzalez-Guillen, and O. Guhne. Geometric measure of entanglement for symmetric
states. Phys. Rev. A., 80:032324, 2009.
C. R. Johnson and R. Reams. Scaling of symmetric matrices by positive diagonal congruence. Linear Multilinear Algebra,
57:123–140, 2009.
M. Marcus. Subpermanents. Amer. Math. Monthly, 76:530–533, 1969.
A. W. Marshall and I. Olkin. Scaling of matrices to achieve specified row and column sums. Numer. Math., 12(1):83–90,
1968.
H. Minc. Permanents. Addison-Wesley Publishing Co., 1978.
R. Pereira. Differentiators and the geometry of polynomials. Journal of Mathematical Analysis and Applications,
285(1):336–348, 2003.
A. Pinkus. Interpolation by matrices. Electron. J. Linear Algebra, 11:281–291, 2004.
A. Shimony. Degree of entanglement. In D.M. Greenberger and A. Zeilinger, editors, Fundamental problems in quantum
theory. A conference held in honor of Professor John A. Wheeler. Proceedings of the conference held in Baltimore, MD,
June 18–22, 1994, volume 755 of Annals of the New York Academy of Sciences, pages 675–679, New York, 1995. New York
Academy of Sciences.
R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical
Statistics, 35(2):876–879, 1964.
T. Wei and P.M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum
states. Phys Rev. A., 68:042307, 2003.
Unauthenticated
Download Date | 6/18/17 11:13 PM