PHYSICAL REVIEW A, VOLUME 65, 052302
Properties of entanglement monotones for three-qubit pure states
R. M. Gingrich
California Institute of Technology, Pasadena, California 91125
共Received 3 July 2001; published 15 April 2002兲
Various parametrizations for the orbits under local unitary transformations of three-qubit pure states are
analyzed. The interconvertibility, symmetry properties, parameter ranges, calculability, and behavior under
measurement are looked at. It is shown that the entanglement monotones of any multipartite pure state uniquely
determine the orbit of that state under local unitary transformations. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in Phys. Rev. A 60,
910 共1999兲. A form for such an entanglement monotone is proposed. A theorem is proved that significantly
reduces the number of entanglement monotones that must be looked at to find the maximal probability of
transforming one multipartite state to another.
DOI: 10.1103/PhysRevA.65.052302
PACS number共s兲: 03.67.⫺a, 03.65.Ud, 03.65.Ta
I. INTRODUCTION
Entanglement is at the heart of the studies of quantum
computation and quantum information theory. It is what
separates these studies from their classical counterparts. If
we are to understand what different phenomena occur when
we look at the true quantum mechanical description of nature
as opposed to the approximations of classical mechanics,
then we must understand how the quantum mechanical description differs from the classical description. Entanglement
is a measure of this difference. While entanglement between
two parties is quite well understood 关2–5兴, the entanglement
within a quantum algorithm or in a state shared between
many parties involves multipartite entanglement, which is
just beginning to be understood 关6 – 8兴.
An integral part of the study of entanglement is determining the probability of transforming one pure state into another by local operations and classical communication
共LOCC兲. For two-part systems this problem was solved, or at
least reduced to the problem of finding the eigenvalues of a
Hermitian matrix, by 关4,5兴. For an N⫻M pure state the
Schmidt decomposition tells us that we can write
The proof of this theorem is constructive so we can actually
write down the transformation that gives us 兩 ␾ 典 from 兩 ␺ 典 .
For pure states of more than two parts no such nice theorem
is known. The question of whether two three-qubit pure
states can be transformed into each other with nonzero probability by LOCC has been solved by Dür et al. 关9兴, but just
getting a reasonable upper bound on that probability when it
is nonzero is unsolved. In this paper I attempt to make some
progress toward solving this problem for three-qubit pure
states and hopefully shed some light on how we might solve
it for larger dimensional spaces and more parts.
One way to find P( 兩 ␺ 典 → 兩 ␾ 典 ) is to look at the entanglement monotones E( 兩 ␺ 典 ) for the two states. For the duration
of this paper ‘‘state’’ will refer to a pure state unless explicitly called a mixed state. An entanglement monotone 共EM兲 is
defined as a function that goes from states to positive real
numbers and does not increase under LOCC. As a convention the value of any EM for a separable state is 0. For mixed
and pure states of any dimension and number of parts the
following theorem holds 关10兴:
P 共 ␳ → ␳ ⬘ 兲 ⫽min
n
兩␺典⫽
E
兺 冑␭ ↑i 兩 i 典 兩 i ⬘ 典 ,
共1兲
i⫽1
where the ␭ ↑i are in increasing order, 兺 i ␭ ↑i ⫽1, the 兩 i 典 and
兩 i ⬘ 典 are an orthonormal set of vectors in spaces A and B,
respectively, and n⫽min(N,M). If we define
k
E k 共 兩 ␺ 典 )⫽
兺 ␭ ↑i ,
k⫽1, . . . ,n⫺1,
i⫽1
共2兲
then the highest attainable probability of transforming 兩 ␺ 典 to
兩 ␾ 典 , P( 兩 ␺ 典 → 兩 ␾ 典 ), is given by 关5兴
P 共 兩 ␺ 典 → 兩 ␾ 典 )⫽min
再
冎
E k共 兩 ␺ 典 兲
,1 ,
E k共 兩 ␾ 典 兲
k⫽1, . . . ,n⫺1. 共3兲
1050-2947/2002/65共5兲/052302共7兲/$20.00
E共 ␳ 兲
E共 ␳⬘兲
,
共4兲
where the minimization is taken over the set of all EMs 关10兴.
This can be seen by considering P( ␳ → ␳ ⬘ ) as an EM for ␳ .
The problem is that this minimization is difficult to take
since there is no known way to characterize all the entanglement monotones for multipartite states. We would like a
‘‘minimal set’’ of EMs similar to the E k for the bipartite case
in order to take the minimization.
The situation for three or more parts is somewhat different
than for bipartite pure states. First, generic M ⫻M bipartite
states have a stabilizer 共i.e., the set of unitaries that takes a
state to itself兲 of dimension M ⫺1 isomorphic to U(1) 丢 M ⫺1
while pure states with more parts generically have a discrete
stabilizer. States whose parts are not of the same dimension
may have larger stabilizers but bipartite states are the only
ones that always have a continuous stabilizer. Secondly, the
generalized Schmidt decomposition, however you choose to
generalize it 关11,12兴, has complex coefficients for pure states
65 052302-1
©2002 The American Physical Society
R. M. GINGRICH
PHYSICAL REVIEW A 65 052302
with three or more parts. This implies that generally these
states are not locally unitarily equivalent to their complex
conjugate states 共i.e., the state with each of its coefficients
complex conjugated兲. Also, for bipartite pure states all the
local unitary 共LU兲 invariants can be calculated from the eigenvalues of the reduced density matrices but this does not
hold for more parts. I will go into more detail about LU
invariants in the next section.
The structure of the paper is as follows. In Sec. II the
interconvertibility, behavior under measurement, symmetry
properties, parameter ranges, and calculability of two generalizations of the Schmidt decomposition of Eq. 共1兲 and the
polynomial invariants 共defined below兲 are looked at. In Sec.
III it is shown that the entanglement monotones uniquely
determine the orbit of multipartite pure states and this is used
to show that there must be an EM algebraically independent
of the known EMs. A form for this EM is proposed and
studied. Section IV discusses other monotones that must exist and their properties. Lastly, in Sec. V a theorem is proved
that significantly reduces the number of EMs that must be
minimized over to get P( ␳ → ␳ ⬘ ) of Eq. 共4兲.
II. DECOMPOSITIONS AND INVARIANTS OF THREEQUBIT PURE STATES
Let 兩 ␺ 典 be a multipartite state in H1 丢 H2 ••• 丢 Hn and let
be Krauss operators for an operation on the
(i)
Hilbert space Hi with 兺 k A (i)†
k A k ⫽Ii and Ii the identity
acting on Hi . A 共nonincreasing兲 EM is a real valued function
E( 兩 ␺ 典 ) such that
A (i)
k :Hi →H⬘
i
E 共 兩 ␺ 典 )⭓
兺k p k E
冉
I 1 丢 ••• 丢 A (i)
k 丢 ••• 丢 I n 兩 ␺ 典
for any state 兩 ␺ 典 , operation
冑p k
A (i)
k ,
冊
A. The polynomial invariants
A general polynomial invariant P ␴ , ␶ ( 兩 ␺ 典 ) for a threequbit state of the form
1
兩␺典⫽
t i jk 兩 i jk 典
共7兲
is written as
P ␴ , ␶ 共 兩 ␺ 典 )⫽
兺 t i j k •••t i j k t̄ i j
1 1 1
n n n
1 ␴ (1) k ␶ (1)
••• t̄ i n j ␴ (n) k ␶ (n)
共8兲
where ␴ and ␶ are permutations on n elements, repeated
indices are summed, and t̄ stands for the complex conjugate
of t 关13兴. If one applies a unitary to any of the qubits in 兩 ␺ 典
and explicitly writes out P ␴ , ␶ ( 兩 ␺ 典 ) again it becomes apparent that P ␴ , ␶ ( 兩 ␺ 典 ) is invariant. Of course, any polynomial in
terms of the polynomial invariants P ␴ , ␶ ( 兩 ␺ 典 ) is another polynomial invariant. In fact, it can be shown that all the polynomial invariants are of this form.
We know from 关11兴 that the number of independent polynomial invariants is given by
dim关 C 2 丢 C 2 丢 C 2 兴 ⫺3dim关 SU共 2 兲兴 ⫺dim关 U共 1 兲兴 ⫺1⫽5,
共9兲
where the last ⫺1 is due to the fact that we are using normalized states. The five independent continuous invariants
are
I 1 ⫽ P e,(12) ,
I 2 ⫽ P (12),e ,
共5兲
I 3 ⫽ P (12),(12) ,
and space i where
2
p k ⫽ 储 I 1 丢 ••• 丢 A (i)
k 丢 ••• 丢 I n 兩 ␺ 典 储 .
兺
i, j,k⫽0
共10兲
I 4 ⫽ P (123),(132) ,
共6兲
This definition for pure states is taken from the definition for
a general state in 关10兴. One can always transform a state into
product states and a product state cannot be transformed into
anything but another product state so the value of an EM for
a product state is chosen to be zero and all other states must
have a non-negative value for the EM. Since A (i)
k can be a
unitary operator or the inverse of that operator, Eq. 共5兲 implies that all EMs must be invariant under LU. Hence, a first
step to understanding the EMs is to look at the LU invariants
that parametrize the set of orbits.
There are many ways to find LU invariants for three-qubit
states 关13,12,14,11,6,15,16兴 some of which can be generalized to more parts and larger spaces, but for now I will
concentrate on the three-qubit case. The three sets of invariants I will look at in this section are the polynomial invariants 关13兴, what I will call the diagonalization decomposition
关12兴, and what I will call the maximization decomposition
关11兴.
I 5⫽
冏兺
t i1 j 1k1t i2 j 2k2t i3 j 3k3t i4 j 4k4
冏
2
⫻ ⑀ i1i2⑀ i3i4⑀ j 1 j 2⑀ j 3 j 4⑀ k1i3⑀ k2i4 ,
where ⑀ 00 ⫽ ⑀ 11 ⫽0, ⑀ 01 ⫽⫺ ⑀ 10 ⫽1, and again repeated indices are summed. I 4 is the Kempe invariant 关1兴. If one
writes out I 5 and uses the identity ⑀ i j ⑀ rs ⫽ ␦ ir ␦ js ⫺ ␦ is ␦ jr it
can be shown that I 5 is just the sum and difference of 64
polynomials of the form in Eq. 共8兲. With one more discrete
invariant
I 6 ⫽sgn关 Im共 P (34)(56),(13524) 兲兴 ,
共11兲
the LU orbit of a three-qubit state is determined uniquely
关12,17兴. In this paper I will define sgn关 x 兴 as 1 for nonnegative numbers and ⫺1 otherwise. The polynomial invariants have the advantage of being easy to compute for any
state and the four previously known independent EMs 关6兴 are
the following simple functions of I 1 , I 2 , I 3 , and I 5 :
052302-2
PROPERTIES OF ENTANGLEMENT MONOTONES FOR . . .
PHYSICAL REVIEW A 65 052302
J 4 ⫽ 冑I 5 ,
␶ (AB)C ⫽2 共 1⫺I 1 兲 ,
␶ (AC)B ⫽2 共 1⫺I 2 兲 ,
J 5⫽
共12兲
␶ (BC)A ⫽2 共 1⫺I 3 兲 ,
冉
冊
1 5
4
⫺I 1 ⫺I 2 ⫺I 3 ⫹ I 4 ⫺2 冑I 5 ,
4 3
3
then the coefficients are given by
␶ ABC ⫽2 冑I 5 .
␮⫾
0 ⫽
B. The diagonalization decomposition
The diagonalization decomposition 共DD兲 introduced by
Acin et al. 关12兴 is accomplished by first defining matrices
(T 0 ) j,k ⫽t 0 jk and (T 1 ) j,k ⫽t 1 jk , where the t i jk are given by
Eq. 共7兲. Next find a unitary operation on space A that makes
T 0 singular and unitaries on spaces B and C that make T 0
diagonal. Use the remaining phase freedom to get rid of as
many phases as possible. What is left is a state of the form
␮⫾
i ⫽
cos共 ␾ ⫾ 兲 ⫽
共13兲
where ␮ i ⭓0, ␮ 0 ⫹ ␮ 1 ⫹ ␮ 2 ⫹ ␮ 3 ⫹ ␮ 4 ⫽1, and 0⭐ ␾ ⭐ ␲ .
Note that generically there are two unitaries that will make
T 0 singular but it can be shown that only one will lead to ␾
between 0 and ␲ . If there is another solution, with ␾ between ␲ and 2 ␲ exclusive, it is referred to as the dual state
of 兩 ␺ DD典 . Some nice properties of DD are that there is a one
to one correspondence with the orbits and there are a set of
invertible functions between the parameters of the decomposition and the set of polynomial invariants given above;
namely,
I 1 ⫽1⫺2 ␮ 0 共 ␮ 2 ⫹ ␮ 4 兲 ⫺2⌬,
I 2 ⫽1⫺2 ␮ 0 共 ␮ 3 ⫹ ␮ 4 兲 ⫺2⌬,
I 3 ⫽1⫺2 ␮ 0 共 ␮ 2 ⫹ ␮ 3 ⫹ ␮ 4 兲 ,
共14兲
I 4 ⫽1⫺3 关共 ␮ 2 ⫹ ␮ 3 兲共 ␮ 0 ⫺ ␮ 4 兲 ⫹ ␮ 4 共 1⫺ ␮ 4 兲 ⫺ ␮ 2 ␮ 3 ␮ 0
␮⫾
0
,
i⫽2,3,4,
J 2 ⫹J 3 ⫹J 4
␮⫾
0
共16兲
,
⫾
⫾ ⫾
␮⫾
1 ␮ 4 ⫹ ␮ 2 ␮ 3 ⫺J 1
⫾ ⫾ ⫾
2 冑␮ ⫾
1 ␮2 ␮3 ␮4
,
⫾ ⫾ ⫾
sgn关共 sin共 ␾ ⫾ 兲兴 ⫽I 6 sgn„冑␮ ⫾
1 ␮ 2 ␮ 3 ␮ 4 兵 J 1 ⫺J 2 J 3 ⫺J 4
2
⫻ 关 J 2 ⫹J 3 ⫹J 4 ⫺ 共 ␮ ⫾
0 兲 兴 其 …,
where ⌼⫽(J 4 ⫹J 5 ) 2 ⫺4(J 1 ⫹J 4 )(J 2 ⫹J 4 )(J 3 ⫹J 4 )⭓0. The
⫹ and ⫺ solutions for the coefficients correspond to 兩 ␺ DD典
and its dual state. The inversion of the equations for I i was
done independently in 关17兴. Note that their definition of I 4 is
different from the one in this paper.
Another nice property of the DD is that we can perform
an arbitrary measurement on it in space A and stay in the DD
form. Since any measurement can be broken into a series of
two outcome measurements 关18兴, we can look at the two
outcome measurement A 1 and A 2 where A †1 A 1 ⫹A †2 A 2 ⫽I.
Using the singular value decomposition we can write A i
⫽U i D i V where V does not depend on i because the two
positive Hermitian operators A †1 A 1 and A †2 A 2 sum to the
identity and therefore must be simultaneously diagonalizable. The diagonal matrices D i can be written as
⫹ 共 1⫺ ␮ 0 兲共 ⌬⫺ ␮ 1 ␮ 4 兲兴 ,
D 1⫽
I 5 ⫽4 ␮ 20 ␮ 24 ,
冋 册
x
0
0
y
D 2⫽
,
冋
冑1⫺x 2
0
0
冑1⫺y 2
册
,
共17兲
where 0⭐x,y⭐1 关9兴. Since we are only concerned with
what orbit the outcomes are in we may choose the U i transformation. Also, matrices of the form
I 6 ⫽sgn兵 sin共 ␾ 兲 ␮ 20 冑␮ 1 ␮ 2 ␮ 3 ␮ 4
⫻ 关 ⌬⫺ ␮ 4 共 1⫺2 ␮ 0 ⫹ ␮ 1 兲 ⫺ ␮ 2 ␮ 3 兴 其 ,
冋
where ⌬⫽ ␮ 1 ␮ 4 ⫹ ␮ 2 ␮ 3 ⫺2 冑␮ 1 ␮ 2 ␮ 3 ␮ 4 cos(␾). If we define
1
J 1 ⫽ 共 1⫺I 1 ⫺I 2 ⫹I 3 ⫺2 冑I 5 兲 ,
4
e i␺1
0
0
e i␺2
册
共18兲
,
where ␺ 1 and ␺ 2 are real numbers, commute with the D i
matrices so the most general V can be written as
冋
1
J 2 ⫽ 共 1⫺I 1 ⫹I 2 ⫺I 3 ⫺2 冑I 5 兲 ,
4
1
J 3 ⫽ 共 1⫹I 1 ⫺I 2 ⫺I 3 ⫺2 冑I 5 兲 ,
4
Ji
⫾
␮⫾
1 ⫽1⫺ ␮ 0 ⫺
兩 ␺ DD典 ⫽ 冑␮ 0 兩 000 典 ⫹ 冑␮ 1 e i ␾ 兩 100 典 ⫹ 冑␮ 2 兩 101 典 ⫹ 冑␮ 3 兩 110 典
⫹ 冑␮ 4 兩 111 典 ,
J 4 ⫹5⫾ 冑⌼
,
2 共 J 1 ⫹J 4 兲
共15兲
␣
冑1⫺ ␣ 2 e i ␪
⫺ 冑1⫺ ␣ 2 e ⫺i ␪
␣
where 0⭐ ␣ ⭐1 and ␪ is real. If we choose
052302-3
册
,
共19兲
R. M. GINGRICH
U 1⫽
1
PHYSICAL REVIEW A 65 052302
冋
⫺x 冑1⫺ ␣ 2 e i ␪
y␣
冑␥ x 冑1⫺ ␣ 2 e ⫺i ␪
y␣
册
,
␥ ⫽y 2 ␣ 2 ⫹x 2 共 1⫺ ␣ 2 兲 ,
共20兲
and similarly for U 2 with (x,y) replaced by
( 冑1⫺x 2 , 冑1⫺y 2 ), then in going from 兩 ␺ DD典 to A 1 兩 ␺ DD典 the
DD coefficients undergo the following transformations:
x y ␮0
,
␥
nately, the parameters as they are given above are not in one
to one correspondence with the orbits. While the decomposition is generically unique, there are choices of the parameters within the given ranges that are not the result of the
decomposition. For example, states with a 2 ⫽1/5⫹ ⑀ , b 2
⫽c 2 ⫽d 2 ⫽ f 2 ⫽1/5⫺ ⑀ /4, and any choice of ␾ have
g
2 2
␮ 0→
1
␮ 1 → 兩 e ⫺i ␪ 共 x 2 ⫺y 2 兲 ␣ 冑␮ 0 共 1⫺ ␣ 2 兲 ⫹e i ␾ ␥ 冑␮ 1 兩 2 ,
␥
共21兲
␮ i→ ␮ i␥ ,
i⫽2,3,4,
␾ →arg关 e ⫺i ␪ 共 x 2 ⫺y 2 兲 ␣ 冑␮ 0 共 1⫺ ␣ 2 兲 ⫹e i ␾ ␥ 冑␮ 1 兴 ,
1
冑2
( 兩 0 典 ⫹ 兩 1 典 ),
1
冑2
冊
( 兩 0 典 ⫹ 兩 1 典 ) ⭓a 2
共25兲
I 2 ⫽1⫺2 关共 a 2 ⫹c 2 兲共 b 2 ⫹d 2 兲 ⫹a 2 f 2 兴 ,
I 3 ⫽1⫺2 关共 a 2 ⫹b 2 兲共 c 2 ⫹d 2 兲 ⫹a 2 f 2 兴 ,
共26兲
I 4 ⫽1⫺3 关 a 共 1⫺a 兲 ⫺ 共 b c ⫹b d ⫹c d 兲共 1⫺2a 兲
2
2
2 2
2 2
2 2
2
⫺2b 2 c 2 d 2 ⫺2abcd f 2 cos共 ␾ 兲兴 ,
I 5 ⫽a 2 兩 a f 2 ⫹4bcde i ␾ 兩 2 ,
The maximization decomposition 共MD兲 关11兴 has a somewhat different way of decomposing the three-qubit states.
First we find the states 兩 ␾ A 典 , 兩 ␾ B 典 , and 兩 ␾ C 典 , each defined
up to an overall phase, that maximize
2
共22兲
and apply a unitary such that 兩 ␾ A 典 兩 ␾ B 典 兩 ␾ C 典 becomes 兩 000 典 .
Defining 兩 1 典 , up to an overall phase, as the vector perpendicular to 兩 0 典 , then the derivative of g along 兩 1 典 at the point
兩 000 典 ,
g 共 兩 0 典 ⫹ ⑀ 兩 1 典 , 兩 0 典 , 兩 0 典 )⫺g 共 兩 0 典 , 兩 0 典 , 兩 0 典 )
⑀
⑀ →0
lim
共23兲
must be zero because g( 兩 0 典 , 兩 0 典 , 兩 0 典 ) is a maximum. Since
we still have phase freedom in 兩 0 典 and 兩 1 典 this implies that
具 ␺ 兩 100 典 ⫽0 and similarly for 具 ␺ 兩 010 典 and 具 ␺ 兩 001 典 . Using
the remaining phase freedom in the choice of 兩 0 典 and 兩 1 典 we
can eliminate all but one phase, leaving us with
兩 ␺ MD典 ⫽ae i ␾ 兩 000 典 ⫹b 兩 011 典 ⫹c 兩 101 典
⫹d 兩 110 典 ⫹ f 兩 111 典 ,
冑2
( 兩 0 典 ⫹ 兩 1 典 ),
I 1 ⫽1⫺2 关共 a 2 ⫹d 2 兲共 b 2 ⫹c 2 兲 ⫹a 2 f 2 兴 ,
C. The maximization decomposition
⫽2 Re关 具 ␺ 兩 100 典具 000 兩 ␺ 典 兴
1
for ⑀ ⭐0.014. Hence, these choices of the parameters are not
a result of the decomposition. The true ranges of the parameters that would give a one to one correspondence with the
orbits are as yet unknown.
A nice property of the MD is that it is symmetric in particle exchange. Exchanging the particles is equivalent to exchanging b, c, and d. This makes the permutation properties
of the polynomial invariants easier to see when written in
terms of the MD coefficients. They take the following forms:
and again similarly for A 2 兩 ␺ DD典 . Things become more complicated when ␾ becomes larger than ␲ and we have a dual
solution. In this case we need to transform to the dual state,
which can be quite tedious. It should also be noted that if we
want to put this last form for the DD coefficients into Eqs.
共14兲 the normalization must be taken into account. The normalization will just be the sum of the forms 共21兲 for ␮ 0
through ␮ 4 .
g 共 兩 ␾ A 典 , 兩 ␾ B 典 , 兩 ␾ C 典 )⫽ 储 具 ␺ 円␾ A 典 兩 ␾ B 典 円␾ C 典 储
冉
共24兲
where a 2 ⫹b 2 ⫹c 2 ⫹d 2 ⫹ f 2 ⫽1, 0⭐ ␾ ⭐2 ␲ , 0⭐a,b,c,d, f ,
and b,c,d, f ⭐a. Note that g( 兩 0 A 典 , 兩 0 B 典 , 兩 0 C 典 )⫽a 2 . Unfortu-
I 6 ⫽sgn兵 abcd f 2 sin共 ␾ 兲关 a 2 共 1⫺2a 2 兲共 1⫺2a 2 ⫺ f 2 兲
⫺4b 2 c 2 d 2 ⫺2abcd f 2 cos共 ␾ 兲兴 其 .
It is apparent from these equations that I 1 , I 2 , and I 3 are
symmetric in permutations of particles AB, AC, and BC,
respectively, and I 4 , I 5 , and I 6 are symmetric in any permutation of the particles. Unfortunately, the equations in 共26兲
are not as easy to invert as those in 共14兲. In fact, just calculating the MD coefficients for an arbitrary state is not an easy
task, as it is in the case of the polynomial invariants and the
DD coefficients, since determining the unitaries for the MD
involves maximizing over a six-dimensional space with typically many local maxima.
One more interesting fact about the MD is that 1⫺a 2 is a
nonincreasing EM. We know this because in 关16兴 it is shown
that a function of the form
E k A ,k B ,k C 共 兩 ␺ 典 )⫽ max 储 ⌫ A 丢 ⌫ B 丢 ⌫ C 兩 ␺ 典 储 2 ,
⌫ A ,⌫ B ,⌫ C
共27兲
where ⌫ X is a k X -dimensional projector on system X
⫽A,B,C, is a nondecreasing EM and E 1,1,1 ( 兩 ␺ 典 )⫽a 2 . The
EM 1⫺a 2 can be shown to be independent of ␶ from Eq.
共12兲 by looking at the gradient vectors of ␶ , 1⫺a 2 and N
⫽a 2 ⫹b 2 ⫹c 2 ⫹d 2 ⫹ f 2 , at, for instance, the point a
⫽3, b,c,d, f ⫽1, and ␾ ⫽ ␲ /2. Since the gradient vectors
052302-4
PROPERTIES OF ENTANGLEMENT MONOTONES FOR . . .
PHYSICAL REVIEW A 65 052302
span a six-dimensional space, 1⫺a 2 cannot be written in
terms of the ␶ and N. The problem with using 1⫺a 2 as an
EM is that one needs to find the global maximum of a sixdimensional space with many local maxima to calculate it.
This is a difficult task for most states.
pend on I 4 since the ␶ are invertible functions of I 1 , I 2 , I 3 ,
and I 5 , respectively. The following function fulfills that criterion:
III. FIFTH INDEPENDENT EM
and numerical results suggest that it is an EM. After generating over 300 000 random states and applying a random
operation to each of them the inequality in Eq. 共5兲 was never
violated by ␴ ABC . Also, note that ␴ ABC is symmetric in particle permutations as is ␶ ABC . For the rest of the paper I will
assume that ␴ ABC is an EM. Indeed, it may be that there is a
set of measure zero or perhaps just a very small measure for
which ␴ ABC is not a monotone and my numerical test did not
explore this space, but there must exist some function of the
polynomial invariants which is independent of the ␶ ’s and is
an EM. For it to be useful in improving our upper bound for
P( 兩 ␺ 典 → 兩 ␾ 典 ) there should be pairs of states 兩 ␺ 典 and 兩 ␾ 典
such that
In Sec. II it was shown that all EMs must be invariant
under LU and hence are determined by the orbit of the state.
For three-qubit states this means that EMs are a function of
only the polynomial invariants, DD coefficients, or MD coefficients. In fact, this determination is unique.
Theorem 1. The set of all EMs for any multipartite pure
state 兩 ␺ 典 uniquely determines the orbit of the state.
Proof. Suppose two states 兩 ␺ 典 and 兩 ␾ 典 in H1 丢 H2 丢 •••
丢 Hn have the same values for the EMs but lie in different
orbits. We know by using Eq. 共4兲 that
P 共 兩 ␺ 典 → 兩 ␾ 典 )⫽ P 共 兩 ␾ 典 → 兩 ␺ 典 )⫽1,
共28兲
so 兩 ␺ 典 can be transformed to 兩 ␾ 典 共and vice versa兲 by n-party
LOCC (n-LOCC), with probability 1. Since EMs are nonincreasing with any n-LOCC they must remain constant during
the entire transformation from 兩 ␺ 典 to 兩 ␾ 典 共and vice versa兲.
Also, we know that any two-party EM (2-EMX ) between a
system X⫽A,B, . . . and the rest of the systems thought of
as one 关e.g., between B and (ACD . . . )兴 is also an n-party
EM. This comes from the fact that the set of n-LOCC is a
subset of the 2-LOCCs for any choice of X. Since a 2-EMX is
nonincreasing over 2-LOCC for any value of X then it is also
nonincreasing under n-LOCC and hence it is an EM. In particular, the sum of the lowest k eigenvectors of the reduced
density matrices,
␴ ABC ⫽3⫺ 共 I 1 ⫹I 2 ⫹I 3 兲 I 4 ,
␴ ABC 共 兩 ␺ 典 )
␶共兩␺典)
⬍min
␴ ABC 共 兩 ␾ 典 )
␶ ␶共兩␾典)
共31兲
共32兲
and I have found such states numerically. The largest value
of
␴ ABC 共 兩 ␺ 典 )
␶兩␺典)
⫺min
␴ ABC 共 兩 ␾ 典 )
␶ ␶共兩␾典)
共33兲
that I found in my limited number of examples of was 0.01.
I was able to find examples of states for which
␶ ( 兩 ␺ 典 )/ ␶ ( 兩 ␾ 典 ) is greater than 1 for all ␶ and
␴ ABC ( 兩 ␺ 典 )/ ␴ ABC ( 兩 ␾ 典 ) is less than 1.
k
E Xk 共 兩 ␺ 典 )⫽
兺 ␭ ↑i 关 ␳ X共 兩 ␺ 典 兲 ]
i⫽1
E Xk ( 兩 ␺ 典 )
关i.e., the 2-EMs in Eq. 共2兲兴 must be EMs. So the
must remain unchanged and hence the spectrum of ␳ X is
unchanged during the transformation from 兩 ␺ 典 to 兩 ␾ 典 . In
particular, an operation on space X, given by A 1 and A 2 ,
must be such that
␳X
冉 冊
A i兩 ␺ 典
冑N
⫽U ␳ X 共 兩 ␺ 典 )U † ,
IV. OTHER EMS AND THE DISCRETE INVARIANT
共29兲
共30兲
where N is the normalization. This can always be satisfied
with A i / 冑N a unitary matrix. Since every operation can be
written as a unitary, 兩 ␺ 典 and 兩 ␾ 典 are unitarily equivalent.
This contradicts our original supposition.
䊏
Since we know there are five parameters that determine
the orbit of a three-qubit state then by Theorem 1 there must
be five independent, continuous EMs. To the best of the author’s knowledge the only four known independent continuous EMs that do not require a difficult maximization over a
multidimensional space are the four ␶ EMs defined in Eq.
共12兲. Any candidate for the fifth independent EM must de-
The five independent continuous EMs ␶ (AB)C , ␶ (AC)B ,
␶ (BC)A , ␶ ABC , and ␴ ABC can easily be inverted to find I 1 –I 5
but to completely determine the orbit of a state we must also
have an EM that will give us the value of the discrete invariant I 6 . This is equivalent to finding an EM that is not the
same for a state and it complex conjugate state. Note that
I 1 , . . . ,I 5 and hence ␶ and ␴ ABC do not change when a state
is conjugated but by looking at any of the sets of LU invariants we can see that generically a state is not LU equivalent
to its conjugate. By looking at Eq. 共4兲 we can see that this
implies that there must be EMs that are not the same for the
generic state and its conjugate. It is also easy to see that for
any operation that takes a state 兩 ␺ 典 to its conjugate 兩 ¯␺ 典 with
probability p there is an operation that takes 兩 ¯␺ 典 to 兩 ␺ 典 with
the same probability. So for a generic state 兩 ␺ 典 there must be
an EM that goes down for the operation 兩 ␺ 典 → 兩 ¯␺ 典 and a
similar one that goes down the same amount for 兩 ¯␺ 典 → 兩 ␺ 典 .
Thus EMs of the following form must exist:
052302-5
␷ ⫾ 共 兩 ␺ 典 )⫽
再
␷ ⫹ ␷ ⬘,
⫾I 6 ⫽1,
␷
otherwise,
共34兲
R. M. GINGRICH
PHYSICAL REVIEW A 65 052302
where ␷ and ␷ ⬘ are functions of ␶ (AB)C , ␶ (AC)B ,
␶ (BC)A , ␶ ABC , and ␴ ABC .
Also, from 关9兴 we know that there are two classes of
three-part
entangled
states
共i.e.,
states
with
␶ (AB)C , ␶ (AC)B , ␶ (BC)A ⬎0兲 that can be converted into each
other with some nonzero probability within the class and
zero probability between the classes; namely, the GHZ class
which contains
兩 GHZ典 ⬅
1
冑2
共 兩 000典 ⫹ 兩 111典 )
共35兲
first for the one-dimensional case.
Lemma 1. If f (x) is an f-type function with n⫽1 then
再 冎
f 共x兲
x
⭓min ,1
f共y兲
y
for any x,y ⑀ S.
Proof.
Case 1. For x⭓y from property 2 we know f (x)⭓ f (y)
and hence
f 共x兲
⭓1.
f共y兲
and has nonzero ␶ ABC and the W class which contains
兩 W典 ⬅
1
冑3
共 兩 001典 ⫹ 兩 010典 ⫹ 兩 100典 )
共36兲
Since ␶ (AB)C , ␶ (AC)B , ␶ (BC)A , ␶ ABC , ␴ ABC , and ␷ ⫾
determine the orbit of the state all other EMs must depend on
them. A fairly general way to create further EMs from known
EMs is to use what I will call f-type functions.
Definition 1. A function f :S傺Rn →R is an f-type function
if it satisfies the following, 共1兲 f (0ជ )⫽0; 共2兲 if x i ⭓y i for all
i⫽1,2, . . . ,n then f (xជ )⭓ f (yជ ) for xជ ,yជ ⑀ S; 共3兲 f „pxជ ⫹(1
⫺p)yជ …⭓p f (xជ )⫹(1⫺ p) f (yជ ) for any xជ ,yជ ⑀ S and 0⭐ p⭐1.
For a set of EMs, 兵 E i 其 , we have
E i 共 兩 ␺ 典 )⭓pE i
冉 冊
A 1兩 ␺ 典
冑p
⫹ 共 1⫺ p 兲 E i
冉 冊
A 2兩 ␺ 典
f 共x兲 x
⭓ . 䊏
f共y兲 y
再 冎
xi
f 共 xជ 兲
⭓min ,1 ,
yi
f 共 yជ 兲
冠 冉 冊
冠 冉 冊冡
ជ
⭓p f E
A 1兩 ␺ 典
冑p
A 1兩 ␺ 典
冑p
ជ
⫹ 共 1⫺ p 兲 E
冉 冊冡
冠 冉 冊冡
冑1⫺ p
ជ
⫹ 共 1⫺ p 兲 f E
冑1⫺ p
,
c⫽min
再冎
xi
.
yi
共44兲
Then we have two cases.
Case 1. If c⭓1 then from property 2 f (xជ )⭓ f (yជ ) and so
f 共 xជ 兲
⭓1.
f 共 yជ 兲
共45兲
Case 2. If c⬍1 then define
z i⫽
xi
,
c
i⫽1,2, . . . ,n,
g共 c 兲
⭓c,
g共 1 兲
共38兲
共46兲
共47兲
or substituting in f we have
where the first inequality comes from property 2 and the
second comes from property 3. Hence, f (E 1 , . . . ,E m ) is
also an EM. We can show that any EM f (E 1 , . . . ,E m ) that
is an f-type function of monotones E 1 , . . . ,E m does not
modify the upper bound on P( 兩 ␺ 典 → 兩 ␾ 典 ) given by
E i共 兩 ␺ 典 )
;
P 共 兩 ␺ 典 → 兩 ␾ 典 )⭐min
i E i共 兩 ␾ 典 )
共43兲
and g(r)⫽ f (rzជ ). Notice that g(r) is an f-type function with
n⫽1 and hence
A 2兩 ␺ 典
A 2兩 ␺ 典
i⫽1,2, . . . ,n,
for xជ ,yជ ⑀ S.
Proof. Let
for any measurement A 1 , A 2 , and any state 兩 ␺ 典 . So we have
ជ
f „Eជ 共 兩 ␺ 典 )…⭓ f pE
共42兲
For n dimensions we have the following theorem.
Theorem 2. If f (x) is an f-type function then
共37兲
冑1⫺ p
共41兲
Case 2. For x⬍y if we choose p⫽(x/y) ⑀ 关 0,1) then we
know from properties 1 and 3 that f (py)⭓p f (y) and so
and has ␶ ABC ⫽0. Looking again at Eq. 共4兲 we see that ␶ ABC
tells us that P( 兩 ␺ W典 → 兩 ␺ GHZ典 )⫽0 but none of the previously
defined EMs tell us that P( 兩 ␺ GHZ典 → 兩 ␺ W典 )⫽0. Since the
only way to get P( 兩 ␺ GHZ典 → 兩 ␺ W典 )⫽0 is to have an EM that
is finite for GHZ-class states and infinite for W-class states or
zero for GHZ-class states and nonzero for W-class states,
such an EM must exist.
V. FINDING A MINIMAL SET
共40兲
f 共 xជ 兲
⭓c.
f 共 zជ 兲
共48兲
Using z i ⭓y i and property 2 we have
共39兲
052302-6
f 共 xជ 兲
⭓c. 䊏
f 共 yជ 兲
共49兲
PROPERTIES OF ENTANGLEMENT MONOTONES FOR . . .
PHYSICAL REVIEW A 65 052302
For three-qubit states if we take the minimum of
over
E⫽ 兵 ␶ (AB)C , ␶ (AC)B , ␶ (BC)A , ␶ ABC ,
E( 兩 ␺ 典 )/E( 兩 ␾ 典 )
␴ ABC , ␷ ⫾ 其 we are actually taking the minimum over the infinite set of all f-type functions of E. Although from Theorem
1 we know that all EMs must be a function of E it is possible
that there exist EMs that are not f-type functions of E. These
EMs could cause P( 兩 ␺ 典 → 兩 ␾ 典 ) to be lower than the minimum of E( 兩 ␺ 典 )/E( 兩 ␾ 典 ) over E. The EM mentioned at the
end of Sec. IV is an example of such an EM.
VI. CONCLUSIONS AND FURTHER RESEARCH
Theorem 1 along with Theorem 2 implies that there
should be a 共not necessarily finite兲 minimal set of EMs M for
which all EMs for three-qubit states or similarly for any type
of multipartite state are f-type functions of M. I conjecture
that such a minimal set should be simple since the f-type
functions seem to be a rather general way of creating EMs
that are functions of other EMs. The difficult part seems to
be finding the EMs that are minimal and showing that they
are minimal. Using numerical results it seems that ␶ may be
minimal. I looked at functions of ␶ that are almost but not
quite f type such as ␶ 1.01 and numerically tested whether they
are EMs or not. None of them were EMs. I cannot say the
same for ␴ ABC and definitely not for ␷ ⫾ since I do not have
an explicit form for ␷ .
There is further research that may help these problems. If
one could invert the equations in 共26兲 to write
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共2000兲.
a, b, c, d, f , and ␾ in terms of I 1 , . . . ,I 6 , that would
allow us to calculate the EM 1⫺a 2 not to mention find the
ranges for and calculate the values of a,b,c,d, f , and ␾ . The
EM 1⫺a 2 could be used to replace ␴ ABC or perhaps as an
addition to E and may prove more useful than ␴ ABC . As far
as finding the minimal EMs and showing that they are minimal, the arbitrary measurement on the DD at the end of Sec.
II B may be useful since it allows us to look at the value of
I 1 , . . . ,I 6 before and after an arbitrary measurement on an
arbitrary state with far fewer parameters than if we did not
take out the LU freedom. Also, it may be able to tell us the
maximal probability of transforming the general complex
state 兩 ␺ 典 to its conjugate state 兩 ¯␺ 典 and this is a crucial piece
of information that is needed to calculate ␷ ⬘ in Eq. 共34兲.
Unfortunately, most of these tasks involve trying to solve
nontrivial equations or systems of equations with many variables, which can be difficult or even impossible.
ACKNOWLEDGMENTS
I would like to thank John Preskill for supporting me
during this research and for many helpful discussions. I
would also like to thank Todd Brun, Sumit Daftuar, Julia
Kempe, Michael Nielsen, Federico Spedalieri, Frank Verstraete, Guifre Vidal, Anthony Sudbery, and David Whitehouse
for interesting discussions. I thank S. Daftuar and D. Whitehouse in particular for assistance in proving Theorem 2.
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7932 共2000兲.
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6725 共2001兲.
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052302-7