Hermitian Tensor Product Approximation of Complex Matrices and Separability

Abstract

The approximation of matrices to the sum of tensor products of Hermitian matrices is studied. A minimum decomposition of matrices on tensor space $H_1\otimes H_2$ in terms of the sum of tensor products of Hermitian matrices on $H_1$ and $H_2$ is presented. From this construction the separability of quantum states is discussed. Comment: 16 pages

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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
1 Introduction
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

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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
2 Tensor 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

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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
?
.
3 Hermitian 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