Page 1

arXiv:quant-ph/0603103v1 11 Mar 2006

Hermitian Tensor Product Approximation

of Complex Matrices and Separability

Shao-Ming Feia,b, Naihuan Jingc,d, Bao-Zhi Suna

aDepartment of Mathematics, Capital Normal University, Beijing 100037

bMax-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig

cDepartment of Mathematics, North Carolina State University,

Raleigh, NC 27695

dDepartment of Mathematics, Hubei University, Wuhan, Hubei 430062

Abstract

The approximation of matrices to the sum of tensor products

of Hermitian matrices is studied. A minimum decomposition of

matrices on tensor space H1⊗H2in terms of the sum of tensor

products of Hermitian matrices on H1and H2is presented. From

this construction the separability of quantum states is discussed.

PACS numbers: 03.67.-a, 03.65.Ud, 03.65.Ta

MSC numbers:

Key words: Separability; Matrix; Tensor product decomposition

1Introduction

The quantum entangled states have become one of the key resources in quantum information

processing. The study of quantum teleportation, quantum cryptography, quantum dense

coding, quantum error correction and parallel computation [1, 2, 3] has spurred a flurry of

activities in the investigation of quantum entanglements. Despite the potential applications

of quantum entangled states, the theory of quantum entanglement itself is far from being

satisfied. The separability for bipartite and multipartite quantum mixed states is one of the

important problems in quantum entanglement.

Let H1 (resp. H2) be an m (resp. n)-dimensional complex Hilbert space, with |i?,

i = 1,...,m (resp. |j?, j = 1,...,n), as an orthonormal basis. A bipartite mixed state is said

to be separable if the density matrix can be written as

ρ =

?

i

piρ1

i⊗ ρ2

i,(1)

1

Page 2

where 0 < pi ≤ 1,?

not exist for a generic mixed state ρ [4, 5, 6]. With considerable effort in analyzing the

separability, there have been some (necessary) criteria for separability in recent years, for

instance, the Bell inequalities [7], PPT (positive partial transposition)[8] (which is also

sufficient for the cases 2 × 2 and 2 × 3 bipartite systems [9]), reduction criterion[10, 11],

majorization criterion[12], entanglement witnesses [9] and [13, 14], realignment [15, 16, 17]

and generalized realignment [18], as well as some necessary and sufficient criteria for low

rank density matrices [19, 20, 21] and also for general ones but not operational [9].

ipi = 1, ρ1

iand ρ2

iare rank one density matrices on H1 and H2

respectively.It is a challenge to find a decomposition like (1) or proving that it does

In [22] the minimum distance (in the sense of matrix norm) between a given matrix and

some other matrices with certain rank is studied. In [23] and [24], for a given matrix A, the

minimum of the Frobenius norm like ||A −?

A, i.e. we require Bi and Ci to be Hermitian matrices. By dealing with the Hermitian

condition as higher dimensional real constraints, an explicit construction of general matrices

on H1⊗H2according to the sum of the tensor products of Hermitian matrices as well as real

symmetric matrices on H1⊗H2is presented. The results are generalized to the multipartite

case. The separability problem is discussed in terms of these tensor product expressions.

iBi⊗ Ci||F is investigated. In this paper we

develop the method of Hermitian tensor product approximation for general complex matrix

2Tensor product decomposition in terms of real sym-

metric matrices

We first consider the tensor product decompositions according to real symmetric matrices.

Let A be a given mn×mn real matrix on H1⊗H2. We consider the problem of approximation

of A such that the Frobenius norm

||A −

r

?

i

Bi⊗ Ci||F

(2)

is minimized for some m × m real symmetric matrix Bion H1and n × n real symmetric

matrix Cion H2, i = 1,...,r ∈ I N.

We first introduce some notations. For an m×m block matrix Z with each block Zijof

size n × n, i,j = 1,...,m, the realigned matrix˜Z is defined by

˜Z = [vec(Z11),···,vec(Zm1),···,vec(Z1m),···,vec(Zmm)]t,

where for any m × n matrix T with entries tij, vec(T) is defined to be

vec(T) = [t11,···,tm1,t12,···,tm2,···,t1n,···,tmn]t.

There is also another useful definition of˜Z, (˜Z)ij,kl= (Z)ik,jl. A matrix Z can be expressed

as the tensor product of two matrices V1on H1and V2on H2, Z = V1⊗ V2if and only if

(cf, e.g., [24])˜Z = vec(V1)vec(V2)t, i.e., the rank of˜Z is one, r(˜Z) = 1.

Due to the property of the Frobenius norm, we have

||A −

r

?

i=1

Bi⊗ Ci||F= ||˜A −

r

?

i=1

vec(Bi)vec(Ci)t||F.(3)

2

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The symmetric condition of the matrices Biand Cican be expressed in terms of some real

matrices S1and S2in a form

St

1vec(Bi) = St

2vec(Ci) = 0,i = 1,...,r.(4)

We define Qsto be an m2×m(m−1)

Qsas {11,21,31,...,m1,12,22,32,...,m2,...,mm}, then all the entries of Qsare zero except

those at 21 and 12 (resp. 31 and 13, ...) which are 1 and −1 respectively in the first (resp.

second, ...) column. We simply denote

2

matrix such that, if we arrange the row indices of

Qs= [{e21,−e12};{e31,−e13};...;{em,m−1,−em−1,m}],

where {e21,−e12} is first column of Qs, with 1 and −1 at the 21 and 12 rows respectively;

while {e31,−e13} is second column of Qs, with 1 and −1 at the 31 and 13 rows respectively;

and so on.

Similarly we define Qato be an m2×m(m+1)

Qa= [{e11};{e21,e12};{e31,e13};...;{e22};{e32,e23};{e42,e24};...;{em,m−1,em−1,m},{emm}].

(5)

2

matrix such that

(6)

For m = 2, we have

Qs=

0

1

−1

0

,Qa=

1 0 0

0 1 0

0 1 0

0 0 1

.

S1can then be expressed as, something like QR decomposition,

S1= Qs≡ Q1

?R1

0

?

, (7)

where R1is a full rank

where¯Qsand¯Qaare obtained by normalizing the norm of every column vector of Qsand

Qato be one.

m(m−1)

2

×m(m−1)

2

matrix, Q1is an orthogonal matrix, Q1=?¯Qs¯Qa

?,

For the case m = 2,

Q1=

0100

1

√2

0

1

√2

1

√2

0

0

−1

0

√2

00

01

,R1=

?√2

?

.(8)

S2has a similar QR decomposition with S2= Q2

?R2

0

?

, by replacing the dimension

m with n in (7).

Set

Qt

1˜AQ2=

?ˆA11

ˆA12

ˆA22

ˆA21

?

.(9)

3

Page 4

Suppose the singular value decomposition ofˆA22is given byˆA22=?r

are the eigenvectors of the matrixˆA22ˆA†

i=1

√λiuivt

i, where r

and λi, i = 1,2,...,r, are the rank and eigenvalues ofˆA†

22ˆA22respectively and ui(resp. vi)

22ˆA22). SetˆBi=√λiui,ˆCi= vi.

22(resp.ˆA†

[Theorem 1]. Let A be an mn × mn real matrix on H1⊗ H2, where dim(H1) = m,

dim(H2) = n. The minimum of the Frobenius norm ||A −?r

given by

?

iBi⊗ Ci||F is obtained for

some m × m real symmetric matrix Bion H1and n × n real symmetric matrix Cion H2,

vec(Bi) = Q1

0

ˆBi

?

, vec(Ci) = Q2

?

0

ˆCi

?

. (10)

[Proof]. Set

Qt

1vec(Bi) =

?ˆ bi

ˆBi

?

,Qt

2vec(Ci) =

?ˆ ci

ˆCi

?

.(11)

From (4) and (7) we have

?

Rt

1

0

??ˆ bi

ˆBi

?

= 0,

?

Rt

2

0

??ˆ ci

ˆCi

?

= 0,

which give rise to ˆ bi= ˆ ci= 0, due to the nonsingularity of Rt

1and Rt

2.

From (3), (9) and (11) we obtain

||A −

?????

=

r

?

i=1

Bi⊗ Ci||F= ||Qt

?ˆA11

?ˆA11

?

1˜AQ2−

r

?

0

i=1

Qt

1vec(Bi)vec(Ci)tQ2||F

?????

=

?????

ˆA12

ˆA22

ˆA21

?

?

−

r

?

r

?

F+ ||ˆA21||2

i=1

?

?0

0

ˆBi

??

0

ˆBiˆCt

ˆCt

?????

i

?

?????

F

?????

?????

ˆA12

ˆA22

ˆA21

−

i=1

0

i

??????

F

=

||ˆA11||2

F+ ||ˆA12||2

F+ ||ˆA22−

r

?

i=1

ˆBiˆCt

i||2

F.

From matrix approximation we have thatˆA22=?r

From Theorem 1 we see that if a real symmetric matrix A has a decomposition of tensor

product of real symmetric matrices, thenˆA11=ˆA12=ˆA21= 0. As an example we consider

the Werner state [25],

i=1ˆBiˆCt

iis the singular value decomposi-

tion (SVD) forˆA22, which results in formula (10).

?.

ρw=1 − F

3

I4×4+4F − 1

3

|Ψ−??Ψ−| =

1−F

3

0

0

0

000

0

0

2F+1

6

1−4F

6

0

1−4F

6

2F+1

6

0

1−F

3

,(12)

where |Ψ−? = (|01? − |10?)/√2. State ρw is separable for F ≤ 1/2 and entangled for

4

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1/2 < F ≤ 1. According to the definition of realignment we have

Here the dimension m = n, hence Q2= Q1is given by (8). From (9) we have

˜ ρw=

1−F

3

0

0

2F+1

6

0

0

0

2F+1

6

0

0

1−F

3

1−4F

6

0

0

1−4F

6

0

.

Qt

1˜ ρwQ2=

?

ˆ ρw11

ˆ ρw21

ˆ ρw12

ˆ ρw22

?

=

4F−1

6

0

0

0

00

0

0

1−F

3

0

2F+1

6

2F+1

6

0

1−F

3

1−4F

6

0

.

Therefore ρwis generally not decomposable according to real symmetric matrices because

(ˆ ρw)11= (4F − 1)/6 ?= 0 as long as F ?= 1/4. From the singular value decomposition of

(ˆ ρw)22,

we have:

(ˆ ρw)22=

1−F

3

0

2F+1

6

0

2F+1

6

0

1−F

3

1−4F

6

0

u1= v1=

1

√2

1

0

1

,εu2= v2=

1

√2

−1

0

1

,εu3= v3=

0

1

0

,

with eigenvalues λ1= 1/4, λ2= λ3= (1−4F)2/36 respectively, where ε = (1−4F)/|1−4F|.

From (10) we have vec(B1) =√λ1(1/√2,0,0,1/√2)t, vec(B2) = ε√λ2(−1/√2,0,0,1/√2)t,

vec(B3) = ε√λ3(0,1/√2,1/√2,0)t. Therefore the best real symmetric matrix tensor prod-

uct decomposition is

ρw≈1

4I2×2⊗ I2×2+1 − 4F

12

(σ1⊗ σ1+ σ3⊗ σ3),

where σiare Pauli matrices σ1=

?

0 1

1 0

?

, σ2=

?

0 −i

i0

?

, σ3=

?

1

0 −1

0

?

.

3Hermitian tensor product decomposition of Hermi-

tian matrices

We consider now the tensor product decompositions according to Hermitian matrices. Let

A be a given mn × mn complex matrix on H1⊗ H2. We first consider the problem of

approximation of A such that the Frobenius norm

||A − B ⊗ C||F= ||˜A − vec(B)vec(C)t||F.(13)

5

Page 6

is minimized for some m×m Hermitian matrix B on H1and n×n Hermitian matrix C on

H2.

In order to impose the Hermitian condition of the matrices B and C, we separate the

matrices B and C into real and imaginary parts such that B = b + iB, C = c + iC, where

b and c (resp. B and C) are the real (resp. imaginary) parts of B and C respectively. As

vec(B) = vec(b) + ivec(B), we have

vec(B)vec(C)t= (vec(b)vec(c)t− vec(B)vec(C)t) + i(vec(b)vec(C)t+ vec(B)vec(c)t).

We now map the complex matrix A to be a real one:

A −→

?

a

A

a

−A

?

,

where a and A are the real and the imaginary parts of A respectively. Now the approx-

imation problem of complex matrices to the tensor product of two Hermitian matrices is

reduced to the problem of real matrices and the results in [23, 24] can be used accordingly.

The problem to minimize ||˜A − vec(B)vec(C)t||F is reduced to minimize

?????

antisymmetric. This condition can be expressed in terms of some real matrices S1and S2

in a form

St

1

±vec(B)

[Lemma 1]. Condition (15) has a QR decomposition such that

?????

?

˜ a

−˜ A

˜ A

˜ a

?

−

?

vec(b)

−vec(B) vec(b)

vec(B)

??vec(c) −vec(C)

vec(C)vec(c)

?t?????

?????

F

(14)

under the Hermitian condition: B = B†, C = C†, i.e., b and c are symmetric, B and C are

?

vec(b)

?

= St

2

?

vec(c)

±vec(C)

?

= 0.(15)

S1= Q1

?

R1

0

?

,S2= Q2

?

R2

0

?

,(16)

where R1and R2are full rank matrices, Q1and Q2are orthogonal matrices.

[Proof]. S1can be generally expressed as

S1=

?

Qs

0

0

Qa

?

,

where Qsand Qaare given by (5) and (6) respectively. The QR decomposition of S1 is

given by

?¯Qs

with¯Qs and¯Qa given in section 2, X1 (resp. Y1) is an m2× m2matrix with the first

m(m−1)/2 (resp. last m(m+1)/2) columns replaced by the matrix¯Qs(resp.¯Qa) and the

rest entries zero, R1is a diagonal matrix with diagonal elements either 1 or

case m = 2,

Q1=

00

¯Qa

00

¯Qa

¯Qs

?

≡

?X1

Y1

X1

Y1

?

,(17)

√2. For the

X1=

0

1

√2

√2

0

0 0 0

0 0 0

0 0 0

0 0 0

−1

,Y1=

0 1

0 0

0 0

0 0

0

1

√2

1

√2

0

0

0

0

1

,R1=

√2 0

0

0

0

0

0

0

01

0

0

√2 0

01

. (18)

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S2has a similar QR decomposition with

Q2=

?X2

Y2

X2

Y2

?

, (19)

by replacing the dimension m with n in the expression of S1.

?

Set

Qt

1

?

?

vec(b)

−vec(B)

vec(c)

−vec(C)

?

?

≡

?

?

ˆ b

−ˆB

ˆ c

−ˆC

?

?

,Qt

1

?

?vec(c)

vec(b)

vec(B)

?

?

≡

?ˇ b

?ˇ c

ˇB

?

,

Qt

2

≡

,Qt

2

vec(C)

≡

ˇC

?

.

(20)

From (15) and (16) we have

?

?

Rt

1

0

??

??

ˆ b

−ˆB

ˆ c

−ˆC

?

?

= 0,

?

?

Rt

1

0

??ˇ b

??ˇ c

ˇB

?

?

= 0,

Rt

2

0

= 0,

Rt

2

0

ˇC

= 0,

which give rise to ˆ b = ˇ b = ˆ c = ˇ c = 0, due to the nonsingularity of Rt

1and Rt

2.

Therefore we have

?

?

vec(b)

−vec(B)

vec(b)

vec(B)

?

= Q1

?

ˆ b

−ˆB

?

?

=

?X1

X1

Y1

Y1

X1

??

Y1

??

0

ˇB

0

−ˆB

?

?

?

=

?

−Y1ˆB

−X1ˆB

?

?

,

?

= Q1

?ˇ b

ˇB

=

?

Y1

X1

=

Y1ˇB

X1ˇB

.

(21)

Thus −Y1ˆB = Y1ˇB, X1ˆB = X1ˇB and

ˇB = (X1+ Y1)−1(X1− Y1)ˆB =

¯Qt

s

¯Qt

a

?¯Qs −¯Qa

?ˆB = Im

s,aˆB,(22)

where Im

identity matrix.

s,a= diag(Im

s,−Im

a), Im

s(resp. Im

a) is an m(m−1)/2 (resp. m(m+1)/2) dimensional

Let P denote the permutation matrix,

P =

?

0Im2×m2

0Im2×m2

?

.

It is easily seen that PQ1P = Q1. From the second formula in (21) we have

Qt

1

?vec(B)

vec(b)

?

=

?ˇB

0

?

.

Hence we have

Qt

1

?

vec(b)

−vec(B) vec(b)

vec(B)

?

=

?

0

ˇB

0−ˆB

?

,(23)

7

Page 8

and, similarly,

Qt

2

?vec(c) −vec(C)

vec(C)vec(c)

?

=

?

0 −ˆC

ˇC0

?

,(24)

where

ˇC = In

s,aˆC,(25)

In

matrix.

s,a= diag(In

s,−In

a), In

s(resp. In

a) is an n(n− 1)/2 (resp. n(n+ 1)/2) dimensional identity

Set

Qt

1

?

˜ a

−˜ A

˜ A

˜ a

?

Q2≡

?ˆA11

ˆA12

ˆA22

ˆA21

?

.(26)

That is

ˆA11= Xt

ˆA12= Xt

ˆA21= Yt

ˆA22= Yt

1˜ aX2+ Yt

1˜ aY2+ Yt

1˜ aX2+ Xt

1˜ aY2+ Xt

1˜ aY2+ Xt

1˜ aX2+ Xt

1˜ aY2+ Yt

1˜ aX2+ Yt

1˜ AY2− Yt

1˜ AX2− Yt

1˜ AY2− Xt

1˜ AX2− Xt

1˜ AX2,

1˜ AY2,

1˜ AX2,

1˜ AY2.

[Theorem 2]. To minimize (13) is equivalent to minimize the following formula

?

||ˆA22+ˆBˇCt||2

F+ ||ˆA11+ˇBˆCt||2

F+ ||ˆA12||2

F+ ||ˆA21||2

F.(27)

[Proof]. From (14), (23) and (24) we obtain

?????

=

?????

?

˜ a

−˜ A

?????Qt

˜ A

˜ a

?

−

?

vec(b)

−vec(B) vec(b)

?

?

?

F+ ||ˆA11+ˇBˆCt||2

vec(B)

??

vec(c) −vec(C)

vec(C)

vec(B)

vec(c)

?t?????

?????

vec(c)

F

?????

1

?

ˆA11

−ˆA21

ˆA11

−ˆA21

||ˆA22+ˆBˇCt||2

˜ a

−˜ A

˜ A

˜ a

Q2− Qt

1

?

vec(b)

−vec(B) vec(b)

ˇB

0

ˇC

?−ˇBˆCt

F+ ||ˆA12||2

??vec(c) −vec(C)

?????

vec(C)

?t

Q2

?????

?????

F

=

????? ?????

?

?

ˆA12

ˆA22

ˆA21

ˆA22

−

?

0

−ˆB

??

0

−ˆBˇCt

0 −ˆC

0

?t?????

F

=

????

????

−

0

?????

????

F

=

?

F+ ||ˆA21||2

F.

?

[Lemma 2]. If the matrix A = a + iA is Hermitian, we have the relations:

ˆA12=ˆA21= 0,

ˆA11= Im

s,aˆA22In

s,a.

[Proof]. As the matrix A = a + iA is Hermitian, i.e., a is symmetric and A is antisym-

metric, we have

(˜ a)ij,kl= (a)ik,jl= (a)jl,ik= (˜ a)ji,lk,(˜ A)ij,kl= (A)ik,jl= −(A)jl,ik= −(˜ A)ji,lk.

8

Page 9

Noting that in our construction of Xαand Yα, (Xα)ij,kl= −(Xα)ji,kl, (Yα)ij,kl= (Yα)ji,kl,

α = 1,2, we obtain

(Xt

1˜ aY2)ij,pq= (X1)kl,ij(˜ a)kl,mn(Y2)mn,pq= −(Xt

1˜ aY2)ij,pq= 0.

Similarly we have Yt

1˜ aX2= Xt

1˜ AX2= Yt

1˜ AY2= 0. HenceˆA12=ˆA21= 0.

s,aX1= X1and Y2In

1˜ aY2+ Xt

s,aˆA22In

From the relations Im

s,a(Yt

Xt

s,aY1= −Y1, Im

s,a= Yt

s,a= −Y2, In

1˜ AX2− Xt

s,aˆA11In

s,aX2= X2, we have

1˜ AY2)In

Im

1˜ aY2+ Xt

1˜ AY2). ThereforeˆA11= Im

For Hermitian matrix A, by using (22), (25) and Lemma 2 we have ||ˆA11+ˇBˆCt||F =

||ˆA11+ Im

to minimize ||A−B⊗C||F(13) is equivalent to minimize ||ˆA22+ˆBˇCt||F, which maybe zero,

whenˆA22= −ˆBˇCt.

Now the minimum of the Frobenius norm ||A −?r

i = 1,2,...,r, are the rank and eigenvalues ofˆA†

eigenvectors of the matrixˆA22ˆA†

results in [23, 24], for Hermitian matrix A the minimum of ||A−?r

[Theorem 3]. Let A be an mn×mn Hermitian matrix on H1⊗H2, where dim(H1) = m,

dim(H2) = n. The minimum of the Frobenius norm ||A−?r

whereˆA22is defined by (26), Bi= bi+ iBi, Ci= ci+ iCi, are given by the relations

?

1˜ aX2)In

1˜ aX2and Im

s,aorˆA22= Im

s,a(Yt

s,a= −(Yt

1˜ AX2−

s,a.

?

s,aˆBˇCtIn

s,a||F= ||Im

s,aˆA11In

s,a+ˆBˇCt||F= ||ˆA22+ˆBˇCt||F. From Lemma 2 we have that

iBi⊗ Ci||F can be obtained readily.

i=1

22ˆA22respectively and ui(resp. vi) are the

22ˆA22). SetˆBi=√λiui,ˇCi= −vi. By using the

i=1Bi⊗Ci||Fis obtained

Suppose the singular value decomposition ofˆA22isˆA22=?r

√λiuivt

i, where r and λi,

22(resp.ˆA†

whenˆA22= −?r

i=1ˆBiˇCt

i.

iBi⊗Ci||Fis obtained for some

m×m Hermitian matrix B on H1and n×n Hermitian matrix C on H2, ifˆA22= −?r

?

i=1ˆBiˇCt

i,

vec(bi)

−vec(Bi)

= Q1

?

0

−ˆBi

?

,

?

vec(ci)

vec(Ci)

?

= Q2

?

0

ˇCi

?

.(28)

As an example we consider the bound entangled state on 2 × 4 (m = 2, n = 4) [26],

b 0 0 0

0 b 0 0

0 0 b 0

0 0 0 b

0 0 0 0

b 0 0 0

0 b 0 0

0 0 b 0

ρb=

1

7b + 1

0

0

0

0

b 0

0 b

0 0

0 0

0 0

b 0

0 b

0 0

0

0

b

0

1+b

2

0

0

√1−b2

2

√1−b2

2

0

0

1+b

2

,(29)

where 0 < b < 1. ρbis a PPT but entangled state. The QR decomposition in our case only

depends the dimensions. Q1is still given by (17) and (18). Q2is a 32×32 matrix with X2,

Y2in (19) given by X2= (f1,f2,f3,f4,f5,f6,010), Y2= (06,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10),

9

Page 10

where siand aiare 16 × 1 column vectors:

f1= (0,1/√2,0,0,−1/√2,0,0,0,0,0,0,0,0,0,0,0)t,

f2= (0,0,1/√2,0,0,0,0,0,−1/√2,0,0,0,0,0,0,0)t,

f3= (0,0,0,1/√2,0,0,0,0,0,0,0,0,−1/√2,0,0,0)t,

f4= (0,0,0,0,0,0,1/√2,0,0,−1/√2,0,0,0,0,0,0)t,

f5= (0,0,0,0,0,0,0,1/√2,0,0,0,0,0,−1/√2,0,0)t,

f6= (0,0,0,0,0,0,0,0,0,0,0,1/√2,0,0,−1/√2,0)t,

a1= (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)t,

a2= (0,1/√2,0,0,1/√2,0,0,0,0,0,0,0,0,0,0,0)t,

a3= (0,0,1/√2,0,0,0,0,0,1/√2,0,0,0,0,0,0,0)t,

a4= (0,0,0,1/√2,0,0,0,0,0,0,0,0,1/√2,0,0,0)t,

a5= (0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)t,

a6= (0,0,0,0,0,0,1/√2,0,0,1/√2,0,0,0,0,0,0)t,

a7= (0,0,0,0,0,0,0,1/√2,0,0,0,0,0,1/√2,0,0)t,

a8= (0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)t,

a9= (0,0,0,0,0,0,0,0,0,0,0,1/√2,0,0,1/√2,0)t,

a10= (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)t,

06and 010are 16 × 6 and 16 × 10 null matrices. From (26) we have

?

where

Qt

1

˜ ρb

0

0

˜ ρb

?

Q2≡

?ˆA11

0

0

ˆA22

?

,

ˆA11=ˆA22=

1

1 + 7b

b 0 0 b 0 b

0 0 0 0 0 0

0 0 0 0 0 0

0

b

0

0 0

0 0

b 0

0

0

0

0 0 0 0 0

b 0 0 b 0

0 b 0 0 b

0

b

0

0 0 0 0 0 0

1+b

2

0 0

?

1−b2

2

b 0 0 b 0

1+b

2

.

From the singular value decomposition ofˆA11we have

ˆB1=

ˆB2=

√3b

1+7b(0,0,1,0)t,

√3b

1+7b(1,0,0,0)t,

ˆB3=

ˆB4=

√

√

λ−

(1+7b)√

1+D2

+(0,D+,0,1)t,

λ+

(1+7b)√

1+D2

−(0,D−,0,1)t,

and

ˇC1= −1

ˇC2= −1

ˇC3= −

ˇC4= −

√3(0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0)t,

√3(1,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0)t,

1

√

λ−(1+D2

√

λ+(1+D2

+)(0,0,0,0,0,0,bD++1+b

2,0,0,

?

?

1−b2

2,b(1 + D+),0,0,b(1 + D+),0,bD++1+b

2)t,

1

−)(0,0,0,0,0,0,bD−+1+b

2,0,0,

1−b2

2,b(1 + D−),0,0,b(1 + D−),0,bD−+1+b

2)t,

where

λ±=1 + b + 6b2±√1 + 2b + b2+ 20b3+ 40b4

2

,

10

Page 11

D±= −1 + b − 2b2±√1 + 2b + b2+ 20b3+ 40b4

2b(1 + 3b)

.

Using (28) we have

ρb =

b

2(1 + 7b)

?0 1

1 0

?

⊗

0 1 0 0

1 0 1 0

0 1 0 1

0 0 1 0

1+b

2+ bD+

0

0

1−b2

2

+

b

2(1 + 7b)

?0 −i

i0

?

⊗

0

0

0i

0

0

i

0

0

0

i

−i

0

0

−i

0−i 0

1−b2

2

0

0

2+ bD+

+

1

(1 + 7b)(1 + D2

+)

?D+ 0

01

?

⊗

0

b(1 + D+)

0

0

b(1 + D+)

0

1+b

+

1

(1 + 7b)(1 + D2

−)

?

D− 0

01

?

⊗

1+b

2+ bD−

0

0

1−b2

2

00

0

1−b2

2

0

0

b(1 + D−)

0

0

b(1 + D−)

0

1+b

2+ bD−

(30)

.

4 Separability of bipartite mixed states

In this section we discuss some properties related to the Hermitian tensor product decom-

position that could give rise to some hints to the separability problem of bipartite mixed

states. From Theorem 3 we can always calculate the tensor product decomposition in terms

of Hermitian matrices for a given density matrix A, A =?r

defined on the subspaces H1and H2. Hence one can not say that A is separable.

i=1Bi⊗ Ci. Nevertheless the

Hermitian matrices Bi and Ciare generally not positive. They are not density matrices

Let m(A) and M(A) denote the smallest and the largest eigenvalues of a Hermitian ma-

trix A. We can transform the decomposition into the one given by another set of Hermitian

11

Page 12

matrices which all have the smallest eigenvalue zero, as follows,

A =

r

?

r

?

+Im⊗

i=1

Bi⊗ Ci=

r

?

i=1

(Bi− m(Bi)Im+ m(Bi)Im) ⊗ (Ci− m(Ci)In+ m(Ci)In)

=

i=1

(Bi− m(Bi)Im) ⊗ (Ci− m(Ci)In) +

r

?

m(Bi)m(Ci)Im⊗ In

i=1

m(Ci)(Bi− m(Bi)Im) ⊗ In

r

?

i=1

m(Bi)(Ci− m(Ci)In) +

r

?

i=1

=

r

?

i=1

(Bi− m(Bi)Im) ⊗ (Ci− m(Ci)In)

+

?

r

?

i=1

m(Ci)(Bi− m(Bi)Im) − m

?

r

?

i=1

m(Ci)(Bi− m(Bi)Im)

?

Im

?

⊗ In

+Im⊗

?

r

?

r

?

i=1

m(Bi)(Ci− m(Ci)In) − m

?

r

?

?

i=1

m(Bi)(Ci− m(Ci)In)

?

In

?

+

?

m

?

i=1

m(Ci)Bi

?

+ m

?

r

?

i=1

m(Bi)Ci

−

r

?

i=1

m(Bi)m(Ci)

?

Im⊗ In,

(31)

where Imand Instand for m × m and n × n identity matrices. The coefficient of Im⊗ In,

?

i=1

qB,C≡ m

r

?

m(Ci)Bi

?

+ m

?

r

?

i=1

m(Bi)Ci

?

−

r

?

i=1

m(Bi)m(Ci)

associated with the decomposition A =?r

qB,Cis decomposition dependent. Associated with another decomposition A =?r′

rability indicator of A, S(A) = max(qB,C) for all possible Hermitian decompositions of A.

With respect to S(A) the associated decomposition is generally of the form

i=1Bi⊗ Ciis not necessary positive.

i=1B′

i⊗

C′

ione would obtain qB′,C′ ?= qB,C. We define the maximum value of qB,C to be the sepa-

A =

?

i

¯Bi⊗¯Ci+ Im⊗¯C +¯B ⊗ In+ S(A)Im⊗ In,(32)

where¯Bi≥ 0,¯Ci≥ 0 ,¯B ≥ 0,¯C ≥ 0 are positive Hermitian matrices.

[Theorem 4]. Let A be a Hermitian positive matrix with tensor product decompositions

of Hermitian matrices like A =?r

S(A) ≤ m(A),

1

2

i=1

−|m(Bi)|(M(Ci) − m(Ci)) − |m(Ci)|(M(Bi) − m(Bi))],

i=1Bi⊗ Ci. A is separable if and only if the separability

indicator S(A) ≥ 0. Moreover S(A) satisfies the following relations:

(33)

S(A) ≥

r

?

[M(Bi)m(Ci) + M(Ci)m(Bi)

(34)

S(A) ≥ m(A) −

?

12

i

M(¯Bi)M(¯Ci).(35)

Page 13

[Proof]. If A is separable, A has a decomposition A =?

??

≥

i

?

If S(A) ≥ 0, then the associated decomposition (32) is already a separable expression of A.

From the decomposition (32) with respect to S(A), we have

iBi⊗ Ciof the form (1), i.e.,

m(Bi) ≥ 0, m(Ci) ≥ 0. We have

S(A) ≥ qB,C

= m

i

m(Ci)Bi

?

+ m

??

m(Bi)m(Ci) −

i

m(Bi)Ci

?

?

−

?

m(Bi)m(Ci)

i

m(Bi)m(Ci)

?

m(Ci)m(Bi) +

?

ii

=

i

m(Bi)m(Ci) ≥ 0.

m(A) ≥

?

i

m(¯Bi⊗¯Ci) + m(¯B) + m(¯C) + S(A) = S(A).

On the other hand we have

S(A) ≥ qB,C≥

?

i

m(m(Ci)Bi) +

?

i

m(m(Bi)Ci) −

+ M(Bi)m(Ci) − |m(Ci)|

?

i

m(Bi)m(Ci)

=

?

+

i

?

m(Bi)m(Ci) + |m(Ci)|

22

?

?

i

?

m(Ci)m(Bi) + |m(Bi)|

2

+ M(Ci)m(Bi) − |m(Bi)|

2

?

−

?

i

m(Bi)m(Ci),

which is just the formula (34).

By using the relations M(B + D) ≤ M(B) + M(D), m(B + D) ≥ m(B) + m(D),

m(B + D) ≤ m(B) + M(D) for any m × m matrices B and D, we have

m(A) = m(?

≤ m(Im⊗¯C +¯B ⊗ In+ S(A)Im⊗ In) + M(?

= m(B) + m(C) + S(A) + M(?

≤ S(A) +?

Hence formula (35) follows.

Generally qB,Cwith respect to our decomposition A =?r

Bi(and Ci) are defined in terms of the singular value decomposition eigenvectors, they are

linear independent. We can choose linear functionals ϕisuch that ϕi(Bj) = δij. Applying

ϕi⊗ 1 to both sides of?r

decomposition A =?r′

r

?

13

i¯Bi⊗¯Ci+ Im⊗¯C +¯B ⊗ In+ S(A)Im⊗ In)

i¯Bi⊗¯Ci)

i¯Bi⊗¯Ci)

iM(¯Bi)M(¯Ci).

?

i=1Bi⊗Cidoes not equal to the

separability indicator S(A). Suppose we have another decomposition A =?r′

i=1B′

i⊗C′

i. As

i=1Bi⊗ Ci =?r′

i=1B′

i=1B′

i⊗ C′

1,...,B′

iwe get Ci =?r′

i=1ϕi(B′

j)C′

j, i.e.,

Ci∈< C′

1,...,C′

r′ >. Similarly we have Bi∈< B′

i⊗C′

in terms of the following transformations

r′ >. Therefore any other Hermitian

ican be obtained from our decomposition A =?r

r

?

i=1Bi⊗Ci

B′

j=

i=1

EijBi,C′

j=

i=1

FijCi,

Page 14

as long as the real matrices E = (Eij) and F = (Fij) satisfy the relation EFt= Ir.

The inequalities (33)-(35) can be served as separability criterion themselves. For in-

stance, if the minimum eigenvalue of A is zero, then A is entangled if the right hand side of

(34) is great than zero.

5Conclusion and remarks

We have developed a method of Hermitian tensor product approximation of general complex

matrices. From which an explicit construction of density matrices on H1⊗ H2in terms of

the sum of tensor products of Hermitian matrices on H1and H2is presented. From this

construction we have shown that a state is separable if and only if the separability indicator

is positive. In principle one can always get a Hermitian tensor product decomposition

of a density matrix by using a basic set of Hermitian matrices. Our approach gives a

decomposition with minimum terms (the number of the terms depends on the rank ofˆA22),

similar to the Schmidt decomposition for bipartite pure states. In example (30) we see that

the 8 × 8 density matrix ρbhas only 4 terms in the tensor product decomposition.

In [27] an entanglement measure called robustness is introduced. For a mixed state ρ

and a separable state ρs, the robustness of ρ relative to ρs, R(ρ||ρs), is the minimal s ≥ 0

for which the density matrix (ρ + sρs)/(1 + s) is separable, i.e. the minimal amount of

mixing with locally prepared states which washes out all entanglement. In particular, the

random robustness of ρ is the one when ρsis taken to be the (separable) identity matrix.

In this case ρ has the form ρ = (1 + t)ρ+

separable if and if the minimum of t is zero. Therefore the separability indicator appeared

from our matrix decompositions is basically the minus of the random robustness, up to

a normalization. Another interesting separable approximations of density matrices was

presented in [28]. This method, so called best separable approximations, was based on

subtracting projections on product vectors from a given density matrix in such a way that

the remainder remained positively defined. In stead expressing a density matrix as the

sum of a separable part and the identity part, this approximation gives rise to a sum of a

separable part and an entangled part from which no more projections on product vectors

can be subtracted.

s− tIm⊗ In/mn), where ρ+

sis separable. ρ is

The results can be generalized to multipartite states. Let’s consider a general l-partite

mixed state ρ1,2,...,lon space H1⊗ H2⊗ ... ⊗ Hl. We can first consider ρ1,2,...,las a bipartite

state on space H1and H2⊗...⊗Hl. By using Theorem 2 we can find the tensor decomposition

ρ1,2,...,l=?r1

∀i, with B2

doing so at last we have the Hermitian tensor product decomposition of the form, ρ1,2,...,l=

?r

Ek= (Ek

i=1B1

i⊗B23...l

i

, where B1

iand B23...l

can be again decomposed as B23...l

being Hermitian matrices on H2and H3⊗ ... ⊗ Hlrespectively. In

i

are Hermitian matrices on H1and H2⊗...⊗Hl

i

respectively. The matrices B23...l

ijand B3...l

i

=?r2

j=1B2

ij⊗ B3...l

ij,

ij

i=1B1

can be obtained, ρ1,2,...,l=?r′

For any given decompositions, in terms of the protocol (31), one has ρ1,2,...,l=?r′

i⊗ B2

i⊗ ... ⊗ Bl

i, where Bk

i=1B′1

iare Hermitian matrices on Hk. New decompositions

i⊗ B′2

ij) are the real matrices satisfying?r′

i⊗...⊗B′l

i⊗ ... ⊗ B′l

j=1E1

i, where B′k

i1jE2

j=?r

i=1Ek

ijBk

i, k = 1,...,l,

i2j...El

ilj= δi1i2δi2i3...δil−1il.

i=1B′1

i, B′2

i⊗

B′2

i+q Id1⊗Id2⊗...⊗Idl, where Idiis the identity matrix on Hi, (B′1

i, ...,

14

Page 15

B′l

matrices. The separability indicator S(ρ1,2,...,l) is the maximal value of the parameter q for all

possible positive Hermitian tensor product decompositions. If the parameter S(ρ1,2,...,l) ≥ 0,

the state ρ1,2,...,lis separable, otherwise it is entangled.

i) are Hermitian matrices such that m(B′k

i) = 0, or part of them (but not all) are identity

Acknowledgments S.M. Fei gratefully acknowledges the warm hospitality of Dept. Math.,

NCSU and the support provided by American Mathematical Society. Jing greatly acknowl-

edges the support from the NSA and Alexander von Humboldt foundation.

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15