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Hermitian tensor product approximation of complex matrices and separability
Vol. 57 (2006)
REPORTS ON MATHEMATICAL PHYSICS
No. 2
HERMITIAN TENSOR P R O D U C T APPROXIMATION OF C O M P L E X MATRICES AND SEPARABILITY SHAO-MING
F E I a'b, N A I H U A N
J I N G c'd
and B A O - Z H I S U N a
a Department of Mathematics, Capital Normal University, Beijing 100037, PR China (e-mail: fei @wiener.iam.uni-bonn.de) b Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany c Department of Mathematics, North Carolina State University, Raleigh, NC 27695 d Department of Mathematics, Hubei University, Wuhan, Hubei 430062 (Received September 5, 2005 - Revised November 25, 2005)
Approximation of matrices to the sum of tensor products of Hermitian matrices is studied. A minimum decomposition of matrices on tensor space H 1 ® / / 2 in terms of the sum of tensor products of Hermitian matrices on H1 and/42 is presented. From this construction the separability of quantum states is discussed. PACS numbers: 03.67.-a, 03.65.Ud, 03.65.Ta. Keywords: separability, matrix, tensor product decomposition.
1.
Introduction
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 [1-3] 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. /-/2) be an m (resp. n)-dimensional complex Hilbert space, with Ii), i = 1 . . . . . m (resp. lJ), J = 1 . . . . . n), as an orthonorrnal basis. A bipartite mixed state is said to be separable if the density matrix can be written as
p = F_, p,p ®
(1)
i
where 0 < Pi < 1, Y~i Pi = 1, p] and p2 are rank-one density matrices on H1 and /42, respectively. It is a challenge to find a decomposition like (1) or to prove that it does not exist for a generic mixed state p [4-6]. With considerable effort in analyzing the separability, there have been found some (necessary) criteria for separability in recent years, for instance, the Bell inequalities [7], PPT (positive [271]
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S . M . FEI, N. JING and B. Z. SUN
partial transposition) [8] (which is also sufficient for the cases 2 x 2 and 2 x 3 bipartite systems [9]), reduction criterion [10, 11], majorization criterion [12], entanglement witnesses [9] and [13, 14], realignment [15-17] and generalized realignment [18], as well as some necessary and sufficient criteria for low rank density matrices [19-21] and also for general ones but not operational [9]. 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 I L A - Y~4 Bi ® Ci [IF is investigated. In this paper we develop the method of Hermitian tensor product approximation for a general complex matrix A, i.e. we require the Bi and Ci should be Hermitian matrices. By dealing with the Hermitian condition as higher dimensional real constraints, an explicit construction of general matrices on H1 ® / / 2 in terms of the sum of tensor products of Hermitian matrices as well as real symmetric matrices on Ha ® / / 2 is presented. The results are generalized to the multipartite case. The separability problem is discussed in terms of these tensor product expressions.
2.
Tensor product decomposition in terms of real symmetric matrices
We first consider the tensor product decompositions in terms of real symmetric matrices. Let A be a given m n x m n real matrix on Ha ® / / 2 . We consider the problem of approximation of A such that the Frobenius norm
Bi ®
A - ~
Ci F
(2)
i
is minimized for some m x m real symmetric matrix Bi o n H1 and some n x n real symmetric matrix Ci on //2, i = 1. . . . . r ~ N. We first introduce some notation. For an m x m block matrix Z with each block Zij of the size n x n, i, j = 1 . . . . . m, the realigned matrix 2 is defined by 2 = [vec(Zll),.-.
, vec(Zml),."
, vec(Zlm),""
, veC(Zmm)] t,
where for any m x n matrix T with entries tij, vec(T) is defined as vec(T) = [tla,-." , tin1, q2, " " , t,n2, " " " , qn, " " " , tmn] t. There is also another useful definition of 2, (Z)ij,kt = (Z)ik,jt. A matrix Z can be expressed as the tensor product of two matrices Va on H1 and V2 on H2, Z = 1/1 ® V2, if and only if (cf. e.g. [24]) 2 = vec(V~)vec(V2) t, i.e. the rank of is one, r ( Z ) = 1. Due to the property of the Frobenius norm we have A - ~ i=a
Bi @ Ci F =
A - ~vec(Bi)vec(Ci) i=1
t F"
(3)
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
The symmetric condition of the matrices Bi and some real matrices S~ and $2 in the form
273
Ci can be expressed in terms of
St1vec(Bi) = S~ vec(Ci) = 0,
i = 1 . . . . . r.
(4)
We define Qs as an m 2 x m(2-1--------~)matrix such that, if we arrange the row indices Qs as {11,21,31 . . . . . m 1 , 1 2 , 2 2 , 3 2 . . . . . m2 . . . . . mm}, then all the entries of Qs are 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 of
Qs =
[{e21, - e l 2 } ; {e31, - e l 3 } ; . . .
;
{em,m-1, --em-l,m}],
(5)
where {e21,-el~} is the first column of Qs, with 1 and - 1 at the 21 and 12 rows, respectively; while {e31,-e13} is the second column of Q~, with 1 and - 1 at the 31 and 13 rows, respectively; and so on. Similarly we define Qa as an m 2 x ~ matrix such that
Qa = [{ell}; {e21, el2}; {e31, el3}; . . . ; {e22}; {e32, e23}; {e42, e24};... ; {era.m-l, ern-l,m}, {em,n}].
[0/ (100)
For m = 2 we have
QS
Qa =
1
(6)
010
-1
010
0
001
$1 can then be expressed as, something like QR decomposition, /
\
$ 1 = Qs = Q1 ( R ~ ] , 0
]
(7)
where R1 is a full rank re(m-l)2 x m(m-1)2 matrix, Q1 is an orthogonal matrix, Q1 = (~)s~)a), where ~)s and ~)a are obtained by normalizing the norm of every column vector of Qs and Qa to one. For the case m = 2, ~
l0 / l/ - ~1 o ~ o 0
0
0
1
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S. M. FEI, N. JING and B. Z. SUN
$2 has a similar QR decomposition with $2 = Q2( R2 ), by replacing the dimension m with n in (7). Set Q]~Q2= ( "~11 /~12) (9)
~i21 A22 Suppose the singular value decomposition of A22 is given by "~22 = ~7=1 ~iUiV~, where r and )~i, i = 1, 2 . . . . . r, are the rank and eigenvalues of A~2A22, respectively, and ui (resp. vi) are the eigenvectors of the matrix AzzA~2 (resp. A~2Az2). Set ~i = ~ / u i ,
d = re.
THEOREM 1. Let A be an m n × m n real matrix on H1 ® H2, where dim(Hi) ---m, dim(H2) = n. The minimum o f the Frobenius norm {IA- ~ Bi ® Cillv is obtained f o r some m x m real symmetric matrix Bi on HI and n × n real symmetric matrix Ci on H2, given by
o)
Proof:
vec(Bi) = Q1 ( ~i
o) " vec(Ci) = Q2 Ci
(lO)
Q~ vec(Bi) = ( ~i
Qt2vec(Ci) = ( di ).
(11)
Set
From (4) and (7) we have (R~ 0 ) ( / }]3i i)=0,
(Ri O ) ( Ci ) = 0 ,
which give rise to fii = ci = 0, due to the nonsingularity of R~ and From (3), (9) and (11) we obtain r
{IN - ~
Bi ® CillF
-~-HQ]AQ2 -
i=1
Rt2,
QtlVeC(Bi)vec(Ci)t Q2llf
i=1
=i
/~21 A22
i=1
°)(o
fi~-21"~22
i=1
0 ~iCLt ) F
F
r
11/~11112 ']-IIA12112 ']-IIA21112 -]-]1/~22 - Z i=1
]~iCi^ ^tII2F.
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
275
From the matrix approximation we have that /~22 ~- ~--~-~=1~id[ is the singular value decomposition (SVD) for J~22, which results in formula (10). [] From Theorem 1 we see that if a real symmetric matrix A has a decomposition as a tensor product of real symmetric matrices, then Jr11 =/~12 =/~21 = 0. AS an example we consider the Werner state [25], 1-F
Pw -- - - I 4 × 4 3
4F-1 -'~- - - I q " - ) < q " - I 3 (1
F)/3
-
0
0
0
0
(2F + 1)/6
(1 - 4 F ) / 6
0
0
(1 - 4 F ) / 6
(2F + 1)/6
0
0
0
0
(1 -
(12)
V)/3
where I ~ - ) = (101)-I10))/v/~. The state Pw is separable for F _< 1/2 and entangled for 1/2 < F _< 1. According to the definition of realignment we have (1 - F)/3
0
0
(2F + 1)/6
0
0
(1 - 4 F ) / 6
0
0
(1 - 4 F ) / 6
0
0
(2F + 1)/6
0
0
(1 - F)/3
Here the dimension m = n, hence
J
Q1 is given by (8). From (9) we have \ (4F - 1)/6 0 0 0
Qtl~wQ2 = (pwll /~w12) = pw21 /0w22
Q2 =
0
(1 - F)/3
0
(2F + 1)/6
0
0
(1 - 4 F ) / 6
0
0
(2F + 1)/6
0
(1 - F)/3
J
Therefore Pw is generally not decomposable in terms of real symmetric matrices because (/3w)11 = ( 4 F - 1)/6 # 0 as long as F # 1/4. From the singular value decomposition of (fiw)22, X (/9w)22 =
we have:
(1 - F)/3
0
(2F + 1)/6
0
(1 - 4 F ) / 6
0
(2F + 1)/6
0
(1 - F)/3
/ '
(1/
276 1 Ul ~
U1 ~--- ~
0
/
S. M. FEI, N. JING and B. Z. SUN
1 ,
6U2 :
1)2 :
0
1
,
6U3 :
1)3 :
(o/
1
1
,
0
with eigenvalues )~1 = 1/4, )~2 = )~3 = ( 1 - 4F)2/36, respectively, where e = ( 1 - 4 F ) / l 1 - 4 F I . From (10) we have vec(B1) = ~/~(1/~/2, 0, 0, 1/~/2) t, vec(B2) = ev/~(-1/V'-2, 0, 0, 1/~/2) ' and vec(B3) = eV~-~(0, 1/~/2, l / Z , 0) t. Therefore the best real symmetric matrix tensor product decomposition is 1 -4F
1
Pw ~ 4-2"I2x2 @ I 2 x 2 ÷ -2- - -1: - -
where
0-i a r e P a u l i
matrices
0-1 :
(o,)
(0-1 ® 0-1 ÷ 0-3 @ O'3),
(o/)
, 0"2 :
0 3.
i
, 0"3 :
0
(,o)
•
0-1
Hermitian tensor product decomposition of Hermitian matrices
We consider now the tensor product decompositions in terms of Hermitian matrices. Let A be a given m n x m n complex matrix on H1 ® H2. We first consider the problem of approximation of A such that the Frobenius norm IIA - B ® CI[F =
IIa -
(13)
vec(B)vec(C)tll•
is minimized for some m x m Hermitian matrix B on H1 and some n x n Hermitian matrix C on /-/2. In order to impose the Hermitian condition on the matrices B and C, we separate the matrices B and C into the real and imaginary parts such that B-----B + i]~, C = c + iC, where B and C (resp. /3 and C) are the real (resp. imaginary) parts of B and C. As v e c ( B ) = v e c ( B ) + i v e c ( B ) , we have v e c ( B ) v e c ( C ) t = (vec(B)vec(C) t -- vec(B)vec(C) r) + i (vec(B)vec(C) t ÷ vec(/3)vec(C)t).
We now map the complex matrix A to a real one, A
> --,,4
A
where A and .A are the real and the imaginary parts of A, respectively. Thus the approximation problem of complex matrices by the tensor product of two Hermitian matrices is reduced to the problem of real matrices and the results of [23, 24] can be used. The problem of minimizing I 1 ' - vec(B)vec(C)tl[F is reduced to minimizing (A -4
.~) _ (vec(B)vec(B))(vec(C) A
--vec(B) vec(B)
vet(C)
-vec(C)) t vec(c)
(14) V
277
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
under the Hermitian condition: B = B t, C = C t, i.e. B and c are symmetric, /3 and C are antisymmetric. This condition can be expressed in terms of some real matrices $1 and $2 in the form
(vector) S~ +vec(/3)
0
= S~ +vec(C)
LEMMA 1. Condition (15) has a QR decomposition such that
S1= QI( R1) ,
$2 = Q2
( 82 ) ,
0
(16)
0
where R1 and R2 are full rank matrices, Q1 and Q2 are orthogonal matrices. Proof: $1 can be generally expressed as
s~ = (Q~ o), 0
Qa
where Qs and Qa are given by (5) and (7) respectively. The QR decomposition of $1 is given by
00aOs 0
Y1X1
with Qs and 0 a given in Section 2, X1 (resp. Y1) is an m e × m 2 matrix with the first m ( m - 1)/2 (resp. last m(m + 1)/2) columns replaced by the matrix Qs (resp. Qa) and the remaining entries zero, R1 is a diagonal matrix with diagonal elements either 1 or ~ . For the case m = 2,
[o ooo/ [OLOO / 1/x/~
x~=
000
-1/C~ooo 0
000
YI=
'
o o 1/~ o oo 1/45o
00
0
,
RI=
1
0
0i)
1 0
o o4~ 0
0
0 (18)
$2 has a similar QR decomposition with
Q2_~_(x2 Y2), I12 X2 by replacing the dimension m with n in the expression for S1.
(19) []
s.M. FEI,N. JINGandB. Z. SUN
278 Set
=(I~/~), Q] (vec(B) -vec(/3) )
Q](vec(B)
13), (20)
=(___~ ) Q~(vec(c) ' -vec(C) ) From (15) and (16) we have
Q~(vec(c) vec(C)) ~ ( ~ ) "
=o,
( < o)
=0,
( R~ 0
=o, ~
=0,
which give rise to fi = g = ~ = c = 0, due to the nonsingularity of R~ and R~. Therefore we have 0 : (-YI~), (vec(B) f3 X 1 Y1 )(_1~) -vec(~)) = Ql(-]3): (Y1 X1 -Xl]3 (21) v (vec(B) Xl Y1 ) ( ~ 0) ( Yl~
vec(/3)) = Q I ( ; ) = ( y
1 X1
:
Xl~)"
Thus -YI~ = YI~, Xl~ : Xl~ and
/3 = (X1 + Y1)-I(x~- Y~)B =
Q~ -Qa )B = I~,aB, m^
~
(22)
I m = diag(Ism -/am), I m (resp. Im) is an m ( m - 1)/2 (resp. m(m + 1)/2)s~a dimensional identity matrix. Let P denote the permutation matrix,
where
P=
( Im2×m2 ° m'm21 0 "
It is easily seen that PQ1P = Q1. From the second formula in (21) we have v
Qt1(vec(/3)
vec(B) ) = (0~)" Hence we have
(vec(B)
vec(/3))
Q] -vec(/3) vec(B) =
( 0 ~) --/~ 0 '
(23)
HERMITIANTENSORPRODUCTAPPROXIMATIONOF COMPLEXMATRICES
279
and, similarly, Q~(vec(c) -vec(C)) vec(C) vec(c) where
(0 -C) d 0
(24)
n ^ = ~,aC,
Isn, a = diag(I~',-/an), I~' (resp. I~) is an identity matrix. Set oi(
n(n-
1)/2 (resp. n(n
+
(25) 1)/2)-dimensional
_~ ~4)Q2~ (All e~12), a
(26,
that is All =
X]AX2 + YfAY2 + X~AY2 - Yf~tX2,
-~12 =
X ] A Y 2 -'1- YfA.X2 -1- X~.J,X2 - Yi'.~,Y2,
A21 = Y[AX2 + X~AY2 + Y[AY2 - x ~ t x 2 ,
-422 = THEOREM 2. Proof:
Y~AY2 + X~AX2 + Y~.4x2 - x~AY2.
The minimization
(13) is
equivalent to minimization of the formula
V/11~22+/~tl]% + ]]An + ~tl]% + 11~12112F+ ]lJ,21H% From (14), (23) and (24) we obtain
( A gee(B) wee(B) wee(C) -gee(C) ) t -A ~)-(-vec(/3) vec(B))(vec(C) vec(C) V
= Q]( A vec(B) vec(B) vec(C) -vec(C) - 4 ~4)Q2-Q~(-vec(/3) vec(B))( vec(C) vec(c)
:
( filll --/~21 = ( /~11 --A21
o o o C ^) t /~12) __ (__~)(~ /~22 0 A21 ) __ ( -~dt 0 A22 0 --HOt / F
(27)
)' Q2
F
[] LEMMA 2.
If the matrix A
= A-}-i.A
/~12 = /~21 = 0,
is Hermitian, we have the relations:
/~11 = Is,aA22Is, m ^ na .
280
s . M . FEI, N. JING and B. Z. SUN
Proof: As the matrix A = A-k-i,A is Hermitian, i.e. A is symmetric and A is antisymmetric, we have (A)ij,kl = (A)ik,jl :
(A)jl,ik = (J~)ji,lk,
(A)ij,kl = (¢4)ik,jl = - - ( A ) j l , i k
= --(A)ji,lk.
Noting that in our construction of X~ and Y~, (X,~)ij,kt = --(X~)ji,kl, (Y,~)ij,kt = (Yc~)ji,kt, v~ = 1, 2, we obtain ( X ~ A Y 2 ) i j , pq : (X1)kl,ij(A)kl,mn(Y2)mn,pq
--(XlAY2)ij,
p q : O.
Similarly we have Y(AX2 = X~fIX2 = Yi'AY2 = 0. Hence A12 = A21 = 0. From the relations I~,aYi = - Y 1 , I~,aX1 = X1 and Y2I~ a = - Y 2 , I~aX2 = X2, we have Isma(Y[aY2 + X] ~.X2)I~, a = Yi' AY2 + X] AX2 and I~,,a(Yi'AX2 - X],AY2)I~, a = - (Y[,A.X2 - X ] A Y z ) . Therefore /~11
m ^ na : I;,aA=I;,
or
A22
: I s , ma A ^l l l s ,. a
[]
For the Hermitian matrix A, by using (22), (25) and Lemma 2 we have [1/~11 "~-13CtllF =
m ^ vt n I 1 / ~ 1 1 - ~ - I ~ , a B C Is,allF :
m ^ [lls,aAllI~n,a -k- ]3dtlIF
:
IIA22+
]3dtHr •
From Lemma 2 we have that minimization of [ [ A - B ® CI[F (13) is equivalent to minimization of l[A22 +BCtlIF, which may be zero when A22 = _/~dt. Now, the minimum of the Frobenius norm I l A - Y~ Bi ®Cil[F can be obtained easily. Suppose the singular value decomposition of A22 is A22 = Y~7=I ~ i u i v ~ , where r and ~i, i . . . .1,. 2, r, are the rank and eigenvalues of A22A2> ^* ^ respectively, and b/i (resp. Vi) are the eigenvectors of the matrix A22/~2 (resp. A~2A22). Set 13i = ~ i u i , Ci = - v i . By using the results of [23, 24], for the Hermitian matrix a the minimum of Ila - ~ = 1 Bi ® CilIF is obtained when A22 = -Z~=I/3iC[THEOREM 3. Let A be an mn x mn Hermitian matrix on H1 ® 142, where dim(Hi) = m, dim(H2) = n. The minimum of the Frobenius norm IIa - Y~ Bi ®Ci IIF is obtained for some m x m Hermitian matrix B on H1 and some n x n Hermitian matrix C on H2, if A22 = - Y~=I 13iC[, where A22 is defined by (26), Bi = Bi +il3i, Ci =- Ci -~ iCi, are given by the relations (vec(Bi)
0 ),
(vec(Ci)
0
(28)
As an example we consider the bound entangled state of the type 2 x 4 (m = 2, n = 4) [26[,
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES (bOO0
0
bO
0
ObO0
0
Ob
0
OObO
0
O0
b
O00b
0
O0
0
1 Pb----
7b+ 1
0 0 0 0 (l+b)/2
0 0 ~-b2/2
bOO0
0
bO
0
ObO0
0
Ob
0
0 0 b 0 ~1-b2/2
281
(29)
0 0 (l+b)/2)
where 0 < b < 1. Pb is a PPT but entangled state. The QR decomposition in our case depends only on dimensions. Q1 is still given by (17) and (18). Q2 is a 32 x 32 matrix with X2, Y2 in (19) given by X2 = (fl, t"2, ['3, f4, 1"5,f6,010), Y2 =
(06,
al, a2, a3, a4, as, a6, a7, a8, a9, alo),
where s i and ai are the 16 × 1 column vectors: fl = (0, 1/V"2, 0, 0, -1/~¢/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) t, t"2 = (0,0, 1/q'2, O, O, O, O, O , - 1 / V ~ , O, O, O, O, O, O, O)t, t"3 = (0, O, O, 1/Vc2, O, O, O, O, O, O, O, O , - 1 / ~ / 2 , O, O, O)t, f4 = ( 0 , 0 , 0 , 0 , 0 , 0 , i / q / 2 , 0 , 0 , - - 1 / ~ / 2 , 0 , 0 , 0 , 0 , 0 , 0 )
t,
fs = (0, O, O, O, O, O, O, l/v/-2, O, O, O, O, O , - 1 / v f 2 , O, O)t, f6 = (0, O, O, O, O, O, O, O, O, O, O, l/x/2, O, O,-1/~v/2, O)t, a 1 ~ ( 1 , 0 , 0 , 0 , 0,0,0, 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 )
t,
a2 ---- (0, 1/~/2, O, O, 1/~f2, O, O, O, O, O, O, O, O, O, O, O)t, a3 = (0,0, 1/~/'2, O, O, O, O, O, 1/~,/2, O, O, O, O, O, O, 0)', a4 = (0, O, O, l/v/2, O, O, O, O, O, O, O, O, 1 / v ~ , O, O, O)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/~v/2, 0, 0, 0, 0, 0, 0) t, a7 = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
1/~/-2, 0, 0, 0, 0, 0, 1/~u/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, alo = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
1) t,
s.M. FEI, N. JING and B. Z. SUN
282
06 and 010 are 16 x 6 and 16 x 10 null matrices. From (26) we have Q] ( fib 0 ) Q2 = ( All
0 ) 0 ~z~22
0 /gb where
fiill = -~22 \
1 1 +7b
bOO
bOb
0
O0
0
00000
0
000
000
b
O0
0
bOObO
b
000
000
0
bO
0
ObOOb
0
000
0 0 0 (l+b)/2
0 0 ~(1-b2)/2
b 0 0 b 0 (l+b)/2
From the singular value decomposition of All we have vf3b
/}l -- 1 + 7-----b(0, 0, 1, 0) t,
/33 =
~2 -- m/~-----~-b(1, 0, 0, 0) t,
~4 ----
dl --
~
D+, 0, 1) t,
(0, n_, 0, 1)t,
(1 + 7b)~/1 + n 2_
1 + 7b
and
~7---(0, (1 + 7b)~/1 + n 2
1
v/~(O, O, O, O, O, O, O, 1, O, O, O, 1, O, O, 1, O)t, 1
Vrj(1, 0, 0, 1,0, 1, O, O, O, O, O, O, O, O, O, O)t, 1
0,0,0,0,0,0,
bD++
1+--2 2 ' o,o,
X//X- (1 + D2+)
.~/-'f_-b2
l + b ) t, b(1 + D+), 0, 0, b(1 + D+), 0, bD+ + - -
V ---~'
d
1
l+b
b 0, 0, 0, 0, 0, 0, bD_ + - - ~ ,
0, 0,
V/X+(1 -4- D2_) ,~_ - b 2 l+b) V - - - 7 ' b(1 + D_), 0, 0, b(1 + D_), O, bD_ + - where
)v-l- ---D+ =
1+ b +
6 b 2 -4- ~/1
+ 2b +
b2
+ 20b 3 +
40b 4
1 + b - 2b 24- ~/1 + 2b + b 2 + 20b 3 + 40b 4 2b(1 + 3b)
t,
J
HERMITIAN TENSOR PRODUCTAPPROXIMATIONOF COMPLEXMATRICES
o,oo] (7 {0i 01)
Using (28) we have
b
Pb--2(1+7b)
/01"~
1010
®
1
b
-i'~
2(1+7b)
0
+
0101
)
®
(o+o) (l+7b)(l+D
2)
0
--Tl+b 0 b D+ ®
q (1 + 7 b ) ( 1 + D2_)
(o_o) 0
0
i
0 -i
0
b(1 + D+)
0
0
b(1 + D+)
0
0
172
1+b2+ bD+]
0
0
1-b 2 2
b(1 + D_)
0
0
0
b(1 + D_)
0
0
0
~-!bD_ ®
0 -i
0
1
~
1
-i
0
0010
q
283
1
172
l+b + 2
bD_
"~
I
(30)
4.
Separability of bipartite mixed states In this section we discuss some properties related to the Hermitian tensor product decomposition that could give 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 = ~ = 1 Bi ® Ci. Nevertheless, the Hermitian matrices Bi and Ci are generally not positive• They are not density matrices defined on the subspaces H1 and //2. Hence one cannot say that A is separable. Let re(A) and M(A) denote the smallest and the largest eigenvalues of a Hermitian matrix A. We can transform the decomposition into the one given by another set of Hermitian matrices which all have the smallest eigenvalue zero, as follows, £ r A = Bi ® Ci = E ( B i - m ( B i ) [ m + m(Bi)Im) ® (Ci - m ( C i ) I n + m ( C i ) I n ) i=l i=l r
r
= ~ . ( 8 i - m(Bi)Im) ® (G - m ( G ) I . ) + ~ m ( G ) ( ~ i - m(~OIm) ® I. i=l i=1 £ r +Im N m(Bi)(Ci - m(Ci)I~) + Em(Bi)m(Ci)Im ® In i=l i=l
S. M. FEi, N. JING and B. Z. SUN
284 r
= ~(Bi
- m(Bi)Im) ® (Ci
i=1
-- m(Ci)In)
I r
+ Zm(Ci)(Bi-m(Bi)Im)-m
i=1 +Im ®
( Zrm ( C i ) ( B i - m ( B i ) I m ) i=1
m(Bi)(Ci - m(Ci)In) - m
i=1
Zm(Bi)(Ci
) Im ] ®In
- m(Ci)In) In
i=1
r
q-[m(i~=l m(fi)ni) -~m(~m(ni)ci) -~m(Bi)m(fi)]lm i=1
(31)
where Im and In stand for an m x m and n x n identity matrices. The coefficient
Of
Im @In,
qB,C -= m
m(Ci)Bi
+m
m(Bi)Ci
-
i=1
m(Bi)m(Ci)
i=1
associated with the decomposition A = }-~=1 Bi @ Ci is not necessary positive. qe,c depends on decomposition. Associated with another decomposition A = Y~f-1 B; ® C~ one would obtain qB',c' 7~ qB,c. We define the maximum value of qB,C to be the separability indicator of A, S ( A ) = max(qs,c) for all possible Hermitian decompositions of A. With respect to S(A) the associated decomposition is generally of the form A = ~
i
[~i ® d i + Im® d + B ® In + S(A)Irn ® 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 the tensor product decompositions into Hermitian matrices like A = ~ = 1 Bi Q Ci. A is separable if and only if the separability indicator S(A) > O. Moreover, S(A) satisfies the following relations: S(A) < m ( a ) , (33)
1 S(A) > ~
1"
Z[M(Bi)m(Ci) + m(Ci)m(Bi) i=1
(34)
- I m ( B i ) l ( M ( C i ) - m(Ci)) - Im(Ci)l(M(Bi) - m(Bi))],
S(A) > re(A) - Z M([~i)M(Ci).
i
(35)
Proof: If A is separable, A has a decomposition A = ~J-~iBi Q Ci of the form (1), i.e. m(Bi) > O, m(Ci) > O. We have
285
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
S(A)>qB,c = m ( ~ m ( C i ) B i ) + m ( ~ i
m(Bi)Ci)-~, m(Bi)m(Ci)
>_ Zm(Ci)m(Bi)..q- ~ m ( B i ) m ( C i ) - Zm(Bi)m(Ci) i
i
i
= Zm(Bi)m(Ci) > O. i If S(A) > O, then the associated decomposition (32) is already a separable expression of A. From the decomposition (32) with respect to S(A) we have
m(A) > Z m(Bi Q Ci) + m([~) + m(C) + S(A) = S(A). i On the other hand we have
S(A) > qB,C >~Zm(m(Ci)Bi) + Zm(m(Bi)Ci) - Zm(Bi)m(Ci) i
i
i
= Z ( m(Bi)m(ci)+lm(Ci)l+ M(Bi)m(Ci)-Im(Ci)J) i 2 2 + Z ( m ( C i ) m(Bi)+lm(Bi)l +M(Ci)m(Bi)21m(Bi)l ) i 2 - Zm(Bi)m(Ci), i which is just the formula (34). By using the relations M(B + D) < M(B) + M(D), m(B + D) > m(B) m(B + D) < m(B) + M(D) for any m × m matrices B and D, we have
m(A) = m ( Z
+ m(D),
[~i ~ Ci "-~ Im ~ C -~- B ~ In q- S ( A ) I m @ In) i
< m(Im ® C + [~ ® In + S(Z)Im ® 11,)+ M( Z [~i ~ Ci) i
=m(B)+m(C)+ S ( A ) + M ( ~ .
Bi@Ci) l
< S(A) + i Hence formula (35) follows.
M([~i)M(Ci). []
r Generally, qB,C with respect to our decomposition A = Y~i=l Bi Q Ci is not equal to the separability indicator S(A). Suppose we have another decomposition A = Y~Tt=l n;~c~. As B i (and Ci) are defined in terms of the singular value decomposition eigenvectors, they are linearly independent. We can choose linear functionals q9i such that qgi(Bj) -~- ~ij" Applying qgi®l to both sides of Zi=lr ni@ci = ZTL1 Bti@Ci: we get
286
Ci
:
s . M . FEI, N. JING and B. Z. SUN F!
Zi=I
~°i (Bj)Cj, t i
i.e. Ci ~ (C 1! .
Cr, . ,, }. Similarly . .
we
. h a v e . Bi E
r!
r
(Brl . . . .
B r')" !
I
Therefore, any other Hermitian decomposition A = ~ i = 1 Bi ® Ci can be obtained r from our decomposition A = ~--~-i=1Bi ® Ci in terms of the following transformations:
Bj :
EijBi, i=l
Cj =
FijCi, i=1
as long as the real matrices E = (Eij) and F = (Fij) satisfy the relation E F t : Ir. The inequalities (33)-(35) can serve as the separability criterion themselves. For instance, if the minimum eigenvalue of A is zero, then A is entangled if the right hand side of (34) is great than zero. 5.
Conclusion and remarks
We have developed a method approximating general complex matrices by Hermitian tensor product. An explicit construction of density matrices on H1 ®/42 in terms of the sum of tensor products of Hermitian matrices on H1 and //2 is presented. Using 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 terms depends on the rank of A22), similar to the Schmidt decomposition for bipartite pure states. In example (30) we see that the 8 × 8 density matrix Pb has only 4 terms in the tensor product decomposition. In [27] an entanglement measure called robustness is introduced. For a mixed state p and a separable state Ps, the robustness of p relative to Ps, R(pllps), is the minimal s >_ 0 for which the density matrix (p + Sps)/(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 p is the one when Ps is taken to be the (separable) identity matrix. In this case p has the form p = (1 + t)p + - t I m ® In/mn), where p+ is separable, p is separable if and only if the minimum of t is zero. Therefore, the separability indicator appearing from our matrix decompositions is basically the minus of the random robustness, up to a normalization. Another interesting separable approximations of density matrices were presented in [28]. This method, the 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. Instead of 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. The results can be generalized to multipartite states. Let us consider a general /-partite mixed state Pl,2,...,l on a space HI ®/42 ® . . . ® Hi. We can first consider Pl,2,...,1 as a bipartite state on the space H1 and //2 ® - . . ®//1. Using Theorem 2 we can find the tensor decomposition Pl,Z,...,t : ~ 1 B/1 ® B/23l, where B/1 and B~ 3"''1
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
287
are Hermitian matrices on H1 and H2 ® . . . @ tit, respectively. The matrices B 23"''1 can be again decomposed as BZi3"•'1= y-~2=1 B~ ® B3ij"''t, ¥ i, with B/} and Bij3...t being Hermitian matrices on H2 and /-/3 @ . . . ® H1 respectively. Doing so we have the Hermitian tensor product decomposition of the form P1,2,...,l = }--~7=1 B] ®B/2®'" .@B~, where B/~ are Hermitian matrices on Hk. New decompositions can be obtained, Pl,e,...,1 . .Y-~;'-I B'~ ® B 'li' where B 'kj = Z i =r I E i jk B i k . .k . = . .1, . . . @ . B'2i @ l,
E k = (E/~) are the real matrices satisfying Y-~}'-I Elt l J .E2. t2J
• . .
E(.tl J = 6ili26i2i3
For any given decomposition, in terms of the protocol (31), one has
3it li~.
• .. _ Pl,2,...,1
=
B']• ® " ® . . . + B i + q Idl ® Id2 ® • • • ® Idt, where Idi is the identity matrix on Hi, (B'], B'~ . . . . . B'ti) are Hermitian matrices such that m(B'ki)= 0, or part of them (but not all) are identity matrices. The separability indicator S ( P l , 2 , . . . , l ) is the maximal value of the parameter q for all possible positive Hermitian tensor product decompositions. If the parameter S(pl,2,...,1) > 0, the state Pl,2,...,l is separable, otherwise it is entangled. Acknowledgements S. M. Fei gratefully acknowledges the warm hospitality of Dept. Math., NCSU and the support provided by American Mathematical Society. N. Jing greatly acknowledges the support from the NSA and Alexander von Humboldt foundation. REFERENCES [1] J. Preskill: The Theory of Quantum Information and Quantum C o m p u t a t i o n , California Inst. o f Tech., 2000, http://www.theory.caltech.edu/people/preskill/ph229/• [2] M.A. Nielsen and I.L. Chuang: Quantum Computation and Quantum I n f o r m a t i o n , Cambridge University Press, Cambridge 2000. [3] D. Bouwmeester, A. Ekert and A. Zeilinger (eds.): The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation and Quantum Computation, Springer, New York 2000. [4] M. Lewenstein, D. Bruss, J. I. Cirac, B. Kraus, M. Kug, J. Samsonowicz, A. Sanpera and R. Tarrach: J. Mod. Opt. 47 (2000), 2841. [5] B. M. Terhal: Theor. Comput. Sci. 287 (2002), 313. [6] M. Horodecki, R Horodecki and R. Horodecki: Springer Tracts in Mod. Phys. 173 (2001), 151. [7] J. S. Bell: Physics (N.Y.) 1 (1964), 195. [8] A. Peres: Phys. Rev. Lett. 77 (1996), 1413. [91 M. Horodecki, E Horodecki and R. Horodecki: Phys. Lett. A 223 (1996), 1. [10] M. Horodecki and R Horodecki: Phys. Rev. A 59 (1999), 4206. [11] N. J. Cerf, C. Adami and R. M. Gingrich: Phys. Rev. A 60 (1999), 898. [12] M. A. Nielsen and J. Kempe: Phys. Rev. Lett. 86 (2001), 5184. [13] B. Terhal: Phys. Lett. A 271 (2000), 319. [14] M. Lewenstein, B. Kraus, J. I. Cirac and R Horodecki: Phys. Rev. A 62 (2000), 052310. [15] O. Rudolph: Physical Review A 67 (2003), 032312. [16] K. Chen and L. A. Wu: Quant. Inf. Comput. 3 (2003), 193. [17] K. Chen and L. A. Wu: Phys. Lett. A 306 (2002), 14. [18] S. Albeverio, K. Chen and S. M. Fei: Phys. Rev. A 68 (2003), 062313. [19] R Horodecki, M. Lewenstein, G. Vidal and I. Cirac: Phys. Rev. A 62 (2000), 032310. [20] S. Albeverio, S. M. Fei and D. Goswami: Phys. Lett. A 286 (2001), 91. [21] S. M. Fei, X. H. Gao, X. H. Wang, Z. X. Wang and K. Wu: Phys. Lett. A 300 (2002), 555.
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[22] G. H. Golub and C. E von Loan: Matrix Computation, The Johns Hopkins University Press, 1989. [23] C. E Van Loan and N. E Pitsianis: in Linear Algebra for Large Scale and Real Time Applications, M. S. Moonen and G. H. Golub (eds.), Kluwer Publications, 1993, pp~ 293-314. [24] N. P. Pitsianis: Ph.D. Thesis, The Kronecker Product in Approximation and Fast Transform Generation, Cornell University, New York 1997. [25] R. E Werner: Phys. Rev. A 40 (1989), 4277. [26] E Horodecki: Phys. Lett. A 232 (1997), 333. [27] G. Vidal and R. Tarrach: Phys. Rev. A 59 (1999), 141. [28] S. Kamas and M. Lewenstein: J. Phys. A 34 (2001), 6919.
REPORTS ON MATHEMATICAL PHYSICS
No. 2
HERMITIAN TENSOR P R O D U C T APPROXIMATION OF C O M P L E X MATRICES AND SEPARABILITY SHAO-MING
F E I a'b, N A I H U A N
J I N G c'd
and B A O - Z H I S U N a
a Department of Mathematics, Capital Normal University, Beijing 100037, PR China (e-mail: fei @wiener.iam.uni-bonn.de) b Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany c Department of Mathematics, North Carolina State University, Raleigh, NC 27695 d Department of Mathematics, Hubei University, Wuhan, Hubei 430062 (Received September 5, 2005 - Revised November 25, 2005)
Approximation of matrices to the sum of tensor products of Hermitian matrices is studied. A minimum decomposition of matrices on tensor space H 1 ® / / 2 in terms of the sum of tensor products of Hermitian matrices on H1 and/42 is presented. From this construction the separability of quantum states is discussed. PACS numbers: 03.67.-a, 03.65.Ud, 03.65.Ta. Keywords: separability, matrix, tensor product decomposition.
1.
Introduction
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 [1-3] 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. /-/2) be an m (resp. n)-dimensional complex Hilbert space, with Ii), i = 1 . . . . . m (resp. lJ), J = 1 . . . . . n), as an orthonorrnal basis. A bipartite mixed state is said to be separable if the density matrix can be written as
p = F_, p,p ®
(1)
i
where 0 < Pi < 1, Y~i Pi = 1, p] and p2 are rank-one density matrices on H1 and /42, respectively. It is a challenge to find a decomposition like (1) or to prove that it does not exist for a generic mixed state p [4-6]. With considerable effort in analyzing the separability, there have been found some (necessary) criteria for separability in recent years, for instance, the Bell inequalities [7], PPT (positive [271]
272
S . M . FEI, N. JING and B. Z. SUN
partial transposition) [8] (which is also sufficient for the cases 2 x 2 and 2 x 3 bipartite systems [9]), reduction criterion [10, 11], majorization criterion [12], entanglement witnesses [9] and [13, 14], realignment [15-17] and generalized realignment [18], as well as some necessary and sufficient criteria for low rank density matrices [19-21] and also for general ones but not operational [9]. 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 I L A - Y~4 Bi ® Ci [IF is investigated. In this paper we develop the method of Hermitian tensor product approximation for a general complex matrix A, i.e. we require the Bi and Ci should be Hermitian matrices. By dealing with the Hermitian condition as higher dimensional real constraints, an explicit construction of general matrices on H1 ® / / 2 in terms of the sum of tensor products of Hermitian matrices as well as real symmetric matrices on Ha ® / / 2 is presented. The results are generalized to the multipartite case. The separability problem is discussed in terms of these tensor product expressions.
2.
Tensor product decomposition in terms of real symmetric matrices
We first consider the tensor product decompositions in terms of real symmetric matrices. Let A be a given m n x m n real matrix on Ha ® / / 2 . We consider the problem of approximation of A such that the Frobenius norm
Bi ®
A - ~
Ci F
(2)
i
is minimized for some m x m real symmetric matrix Bi o n H1 and some n x n real symmetric matrix Ci on //2, i = 1. . . . . r ~ N. We first introduce some notation. For an m x m block matrix Z with each block Zij of the size n x n, i, j = 1 . . . . . m, the realigned matrix 2 is defined by 2 = [vec(Zll),.-.
, vec(Zml),."
, vec(Zlm),""
, veC(Zmm)] t,
where for any m x n matrix T with entries tij, vec(T) is defined as vec(T) = [tla,-." , tin1, q2, " " , t,n2, " " " , qn, " " " , tmn] t. There is also another useful definition of 2, (Z)ij,kt = (Z)ik,jt. A matrix Z can be expressed as the tensor product of two matrices Va on H1 and V2 on H2, Z = 1/1 ® V2, if and only if (cf. e.g. [24]) 2 = vec(V~)vec(V2) t, i.e. the rank of is one, r ( Z ) = 1. Due to the property of the Frobenius norm we have A - ~ i=a
Bi @ Ci F =
A - ~vec(Bi)vec(Ci) i=1
t F"
(3)
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
The symmetric condition of the matrices Bi and some real matrices S~ and $2 in the form
273
Ci can be expressed in terms of
St1vec(Bi) = S~ vec(Ci) = 0,
i = 1 . . . . . r.
(4)
We define Qs as an m 2 x m(2-1--------~)matrix such that, if we arrange the row indices Qs as {11,21,31 . . . . . m 1 , 1 2 , 2 2 , 3 2 . . . . . m2 . . . . . mm}, then all the entries of Qs are 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 of
Qs =
[{e21, - e l 2 } ; {e31, - e l 3 } ; . . .
;
{em,m-1, --em-l,m}],
(5)
where {e21,-el~} is the first column of Qs, with 1 and - 1 at the 21 and 12 rows, respectively; while {e31,-e13} is the second column of Q~, with 1 and - 1 at the 31 and 13 rows, respectively; and so on. Similarly we define Qa as an m 2 x ~ matrix such that
Qa = [{ell}; {e21, el2}; {e31, el3}; . . . ; {e22}; {e32, e23}; {e42, e24};... ; {era.m-l, ern-l,m}, {em,n}].
[0/ (100)
For m = 2 we have
QS
Qa =
1
(6)
010
-1
010
0
001
$1 can then be expressed as, something like QR decomposition, /
\
$ 1 = Qs = Q1 ( R ~ ] , 0
]
(7)
where R1 is a full rank re(m-l)2 x m(m-1)2 matrix, Q1 is an orthogonal matrix, Q1 = (~)s~)a), where ~)s and ~)a are obtained by normalizing the norm of every column vector of Qs and Qa to one. For the case m = 2, ~
l0 / l/ - ~1 o ~ o 0
0
0
1
274
S. M. FEI, N. JING and B. Z. SUN
$2 has a similar QR decomposition with $2 = Q2( R2 ), by replacing the dimension m with n in (7). Set Q]~Q2= ( "~11 /~12) (9)
~i21 A22 Suppose the singular value decomposition of A22 is given by "~22 = ~7=1 ~iUiV~, where r and )~i, i = 1, 2 . . . . . r, are the rank and eigenvalues of A~2A22, respectively, and ui (resp. vi) are the eigenvectors of the matrix AzzA~2 (resp. A~2Az2). Set ~i = ~ / u i ,
d = re.
THEOREM 1. Let A be an m n × m n real matrix on H1 ® H2, where dim(Hi) ---m, dim(H2) = n. The minimum o f the Frobenius norm {IA- ~ Bi ® Cillv is obtained f o r some m x m real symmetric matrix Bi on HI and n × n real symmetric matrix Ci on H2, given by
o)
Proof:
vec(Bi) = Q1 ( ~i
o) " vec(Ci) = Q2 Ci
(lO)
Q~ vec(Bi) = ( ~i
Qt2vec(Ci) = ( di ).
(11)
Set
From (4) and (7) we have (R~ 0 ) ( / }]3i i)=0,
(Ri O ) ( Ci ) = 0 ,
which give rise to fii = ci = 0, due to the nonsingularity of R~ and From (3), (9) and (11) we obtain r
{IN - ~
Bi ® CillF
-~-HQ]AQ2 -
i=1
Rt2,
QtlVeC(Bi)vec(Ci)t Q2llf
i=1
=i
/~21 A22
i=1
°)(o
fi~-21"~22
i=1
0 ~iCLt ) F
F
r
11/~11112 ']-IIA12112 ']-IIA21112 -]-]1/~22 - Z i=1
]~iCi^ ^tII2F.
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
275
From the matrix approximation we have that /~22 ~- ~--~-~=1~id[ is the singular value decomposition (SVD) for J~22, which results in formula (10). [] From Theorem 1 we see that if a real symmetric matrix A has a decomposition as a tensor product of real symmetric matrices, then Jr11 =/~12 =/~21 = 0. AS an example we consider the Werner state [25], 1-F
Pw -- - - I 4 × 4 3
4F-1 -'~- - - I q " - ) < q " - I 3 (1
F)/3
-
0
0
0
0
(2F + 1)/6
(1 - 4 F ) / 6
0
0
(1 - 4 F ) / 6
(2F + 1)/6
0
0
0
0
(1 -
(12)
V)/3
where I ~ - ) = (101)-I10))/v/~. The state Pw is separable for F _< 1/2 and entangled for 1/2 < F _< 1. According to the definition of realignment we have (1 - F)/3
0
0
(2F + 1)/6
0
0
(1 - 4 F ) / 6
0
0
(1 - 4 F ) / 6
0
0
(2F + 1)/6
0
0
(1 - F)/3
Here the dimension m = n, hence
J
Q1 is given by (8). From (9) we have \ (4F - 1)/6 0 0 0
Qtl~wQ2 = (pwll /~w12) = pw21 /0w22
Q2 =
0
(1 - F)/3
0
(2F + 1)/6
0
0
(1 - 4 F ) / 6
0
0
(2F + 1)/6
0
(1 - F)/3
J
Therefore Pw is generally not decomposable in terms of real symmetric matrices because (/3w)11 = ( 4 F - 1)/6 # 0 as long as F # 1/4. From the singular value decomposition of (fiw)22, X (/9w)22 =
we have:
(1 - F)/3
0
(2F + 1)/6
0
(1 - 4 F ) / 6
0
(2F + 1)/6
0
(1 - F)/3
/ '
(1/
276 1 Ul ~
U1 ~--- ~
0
/
S. M. FEI, N. JING and B. Z. SUN
1 ,
6U2 :
1)2 :
0
1
,
6U3 :
1)3 :
(o/
1
1
,
0
with eigenvalues )~1 = 1/4, )~2 = )~3 = ( 1 - 4F)2/36, respectively, where e = ( 1 - 4 F ) / l 1 - 4 F I . From (10) we have vec(B1) = ~/~(1/~/2, 0, 0, 1/~/2) t, vec(B2) = ev/~(-1/V'-2, 0, 0, 1/~/2) ' and vec(B3) = eV~-~(0, 1/~/2, l / Z , 0) t. Therefore the best real symmetric matrix tensor product decomposition is 1 -4F
1
Pw ~ 4-2"I2x2 @ I 2 x 2 ÷ -2- - -1: - -
where
0-i a r e P a u l i
matrices
0-1 :
(o,)
(0-1 ® 0-1 ÷ 0-3 @ O'3),
(o/)
, 0"2 :
0 3.
i
, 0"3 :
0
(,o)
•
0-1
Hermitian tensor product decomposition of Hermitian matrices
We consider now the tensor product decompositions in terms of Hermitian matrices. Let A be a given m n x m n complex matrix on H1 ® H2. We first consider the problem of approximation of A such that the Frobenius norm IIA - B ® CI[F =
IIa -
(13)
vec(B)vec(C)tll•
is minimized for some m x m Hermitian matrix B on H1 and some n x n Hermitian matrix C on /-/2. In order to impose the Hermitian condition on the matrices B and C, we separate the matrices B and C into the real and imaginary parts such that B-----B + i]~, C = c + iC, where B and C (resp. /3 and C) are the real (resp. imaginary) parts of B and C. As v e c ( B ) = v e c ( B ) + i v e c ( B ) , we have v e c ( B ) v e c ( C ) t = (vec(B)vec(C) t -- vec(B)vec(C) r) + i (vec(B)vec(C) t ÷ vec(/3)vec(C)t).
We now map the complex matrix A to a real one, A
> --,,4
A
where A and .A are the real and the imaginary parts of A, respectively. Thus the approximation problem of complex matrices by the tensor product of two Hermitian matrices is reduced to the problem of real matrices and the results of [23, 24] can be used. The problem of minimizing I 1 ' - vec(B)vec(C)tl[F is reduced to minimizing (A -4
.~) _ (vec(B)vec(B))(vec(C) A
--vec(B) vec(B)
vet(C)
-vec(C)) t vec(c)
(14) V
277
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
under the Hermitian condition: B = B t, C = C t, i.e. B and c are symmetric, /3 and C are antisymmetric. This condition can be expressed in terms of some real matrices $1 and $2 in the form
(vector) S~ +vec(/3)
0
= S~ +vec(C)
LEMMA 1. Condition (15) has a QR decomposition such that
S1= QI( R1) ,
$2 = Q2
( 82 ) ,
0
(16)
0
where R1 and R2 are full rank matrices, Q1 and Q2 are orthogonal matrices. Proof: $1 can be generally expressed as
s~ = (Q~ o), 0
Qa
where Qs and Qa are given by (5) and (7) respectively. The QR decomposition of $1 is given by
00aOs 0
Y1X1
with Qs and 0 a given in Section 2, X1 (resp. Y1) is an m e × m 2 matrix with the first m ( m - 1)/2 (resp. last m(m + 1)/2) columns replaced by the matrix Qs (resp. Qa) and the remaining entries zero, R1 is a diagonal matrix with diagonal elements either 1 or ~ . For the case m = 2,
[o ooo/ [OLOO / 1/x/~
x~=
000
-1/C~ooo 0
000
YI=
'
o o 1/~ o oo 1/45o
00
0
,
RI=
1
0
0i)
1 0
o o4~ 0
0
0 (18)
$2 has a similar QR decomposition with
Q2_~_(x2 Y2), I12 X2 by replacing the dimension m with n in the expression for S1.
(19) []
s.M. FEI,N. JINGandB. Z. SUN
278 Set
=(I~/~), Q] (vec(B) -vec(/3) )
Q](vec(B)
13), (20)
=(___~ ) Q~(vec(c) ' -vec(C) ) From (15) and (16) we have
Q~(vec(c) vec(C)) ~ ( ~ ) "
=o,
( < o)
=0,
( R~ 0
=o, ~
=0,
which give rise to fi = g = ~ = c = 0, due to the nonsingularity of R~ and R~. Therefore we have 0 : (-YI~), (vec(B) f3 X 1 Y1 )(_1~) -vec(~)) = Ql(-]3): (Y1 X1 -Xl]3 (21) v (vec(B) Xl Y1 ) ( ~ 0) ( Yl~
vec(/3)) = Q I ( ; ) = ( y
1 X1
:
Xl~)"
Thus -YI~ = YI~, Xl~ : Xl~ and
/3 = (X1 + Y1)-I(x~- Y~)B =
Q~ -Qa )B = I~,aB, m^
~
(22)
I m = diag(Ism -/am), I m (resp. Im) is an m ( m - 1)/2 (resp. m(m + 1)/2)s~a dimensional identity matrix. Let P denote the permutation matrix,
where
P=
( Im2×m2 ° m'm21 0 "
It is easily seen that PQ1P = Q1. From the second formula in (21) we have v
Qt1(vec(/3)
vec(B) ) = (0~)" Hence we have
(vec(B)
vec(/3))
Q] -vec(/3) vec(B) =
( 0 ~) --/~ 0 '
(23)
HERMITIANTENSORPRODUCTAPPROXIMATIONOF COMPLEXMATRICES
279
and, similarly, Q~(vec(c) -vec(C)) vec(C) vec(c) where
(0 -C) d 0
(24)
n ^ = ~,aC,
Isn, a = diag(I~',-/an), I~' (resp. I~) is an identity matrix. Set oi(
n(n-
1)/2 (resp. n(n
+
(25) 1)/2)-dimensional
_~ ~4)Q2~ (All e~12), a
(26,
that is All =
X]AX2 + YfAY2 + X~AY2 - Yf~tX2,
-~12 =
X ] A Y 2 -'1- YfA.X2 -1- X~.J,X2 - Yi'.~,Y2,
A21 = Y[AX2 + X~AY2 + Y[AY2 - x ~ t x 2 ,
-422 = THEOREM 2. Proof:
Y~AY2 + X~AX2 + Y~.4x2 - x~AY2.
The minimization
(13) is
equivalent to minimization of the formula
V/11~22+/~tl]% + ]]An + ~tl]% + 11~12112F+ ]lJ,21H% From (14), (23) and (24) we obtain
( A gee(B) wee(B) wee(C) -gee(C) ) t -A ~)-(-vec(/3) vec(B))(vec(C) vec(C) V
= Q]( A vec(B) vec(B) vec(C) -vec(C) - 4 ~4)Q2-Q~(-vec(/3) vec(B))( vec(C) vec(c)
:
( filll --/~21 = ( /~11 --A21
o o o C ^) t /~12) __ (__~)(~ /~22 0 A21 ) __ ( -~dt 0 A22 0 --HOt / F
(27)
)' Q2
F
[] LEMMA 2.
If the matrix A
= A-}-i.A
/~12 = /~21 = 0,
is Hermitian, we have the relations:
/~11 = Is,aA22Is, m ^ na .
280
s . M . FEI, N. JING and B. Z. SUN
Proof: As the matrix A = A-k-i,A is Hermitian, i.e. A is symmetric and A is antisymmetric, we have (A)ij,kl = (A)ik,jl :
(A)jl,ik = (J~)ji,lk,
(A)ij,kl = (¢4)ik,jl = - - ( A ) j l , i k
= --(A)ji,lk.
Noting that in our construction of X~ and Y~, (X,~)ij,kt = --(X~)ji,kl, (Y,~)ij,kt = (Yc~)ji,kt, v~ = 1, 2, we obtain ( X ~ A Y 2 ) i j , pq : (X1)kl,ij(A)kl,mn(Y2)mn,pq
--(XlAY2)ij,
p q : O.
Similarly we have Y(AX2 = X~fIX2 = Yi'AY2 = 0. Hence A12 = A21 = 0. From the relations I~,aYi = - Y 1 , I~,aX1 = X1 and Y2I~ a = - Y 2 , I~aX2 = X2, we have Isma(Y[aY2 + X] ~.X2)I~, a = Yi' AY2 + X] AX2 and I~,,a(Yi'AX2 - X],AY2)I~, a = - (Y[,A.X2 - X ] A Y z ) . Therefore /~11
m ^ na : I;,aA=I;,
or
A22
: I s , ma A ^l l l s ,. a
[]
For the Hermitian matrix A, by using (22), (25) and Lemma 2 we have [1/~11 "~-13CtllF =
m ^ vt n I 1 / ~ 1 1 - ~ - I ~ , a B C Is,allF :
m ^ [lls,aAllI~n,a -k- ]3dtlIF
:
IIA22+
]3dtHr •
From Lemma 2 we have that minimization of [ [ A - B ® CI[F (13) is equivalent to minimization of l[A22 +BCtlIF, which may be zero when A22 = _/~dt. Now, the minimum of the Frobenius norm I l A - Y~ Bi ®Cil[F can be obtained easily. Suppose the singular value decomposition of A22 is A22 = Y~7=I ~ i u i v ~ , where r and ~i, i . . . .1,. 2, r, are the rank and eigenvalues of A22A2> ^* ^ respectively, and b/i (resp. Vi) are the eigenvectors of the matrix A22/~2 (resp. A~2A22). Set 13i = ~ i u i , Ci = - v i . By using the results of [23, 24], for the Hermitian matrix a the minimum of Ila - ~ = 1 Bi ® CilIF is obtained when A22 = -Z~=I/3iC[THEOREM 3. Let A be an mn x mn Hermitian matrix on H1 ® 142, where dim(Hi) = m, dim(H2) = n. The minimum of the Frobenius norm IIa - Y~ Bi ®Ci IIF is obtained for some m x m Hermitian matrix B on H1 and some n x n Hermitian matrix C on H2, if A22 = - Y~=I 13iC[, where A22 is defined by (26), Bi = Bi +il3i, Ci =- Ci -~ iCi, are given by the relations (vec(Bi)
0 ),
(vec(Ci)
0
(28)
As an example we consider the bound entangled state of the type 2 x 4 (m = 2, n = 4) [26[,
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES (bOO0
0
bO
0
ObO0
0
Ob
0
OObO
0
O0
b
O00b
0
O0
0
1 Pb----
7b+ 1
0 0 0 0 (l+b)/2
0 0 ~-b2/2
bOO0
0
bO
0
ObO0
0
Ob
0
0 0 b 0 ~1-b2/2
281
(29)
0 0 (l+b)/2)
where 0 < b < 1. Pb is a PPT but entangled state. The QR decomposition in our case depends only on dimensions. Q1 is still given by (17) and (18). Q2 is a 32 x 32 matrix with X2, Y2 in (19) given by X2 = (fl, t"2, ['3, f4, 1"5,f6,010), Y2 =
(06,
al, a2, a3, a4, as, a6, a7, a8, a9, alo),
where s i and ai are the 16 × 1 column vectors: fl = (0, 1/V"2, 0, 0, -1/~¢/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) t, t"2 = (0,0, 1/q'2, O, O, O, O, O , - 1 / V ~ , O, O, O, O, O, O, O)t, t"3 = (0, O, O, 1/Vc2, O, O, O, O, O, O, O, O , - 1 / ~ / 2 , O, O, O)t, f4 = ( 0 , 0 , 0 , 0 , 0 , 0 , i / q / 2 , 0 , 0 , - - 1 / ~ / 2 , 0 , 0 , 0 , 0 , 0 , 0 )
t,
fs = (0, O, O, O, O, O, O, l/v/-2, O, O, O, O, O , - 1 / v f 2 , O, O)t, f6 = (0, O, O, O, O, O, O, O, O, O, O, l/x/2, O, O,-1/~v/2, O)t, a 1 ~ ( 1 , 0 , 0 , 0 , 0,0,0, 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 )
t,
a2 ---- (0, 1/~/2, O, O, 1/~f2, O, O, O, O, O, O, O, O, O, O, O)t, a3 = (0,0, 1/~/'2, O, O, O, O, O, 1/~,/2, O, O, O, O, O, O, 0)', a4 = (0, O, O, l/v/2, O, O, O, O, O, O, O, O, 1 / v ~ , O, O, O)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/~v/2, 0, 0, 0, 0, 0, 0) t, a7 = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
1/~/-2, 0, 0, 0, 0, 0, 1/~u/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, alo = ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
1) t,
s.M. FEI, N. JING and B. Z. SUN
282
06 and 010 are 16 x 6 and 16 x 10 null matrices. From (26) we have Q] ( fib 0 ) Q2 = ( All
0 ) 0 ~z~22
0 /gb where
fiill = -~22 \
1 1 +7b
bOO
bOb
0
O0
0
00000
0
000
000
b
O0
0
bOObO
b
000
000
0
bO
0
ObOOb
0
000
0 0 0 (l+b)/2
0 0 ~(1-b2)/2
b 0 0 b 0 (l+b)/2
From the singular value decomposition of All we have vf3b
/}l -- 1 + 7-----b(0, 0, 1, 0) t,
/33 =
~2 -- m/~-----~-b(1, 0, 0, 0) t,
~4 ----
dl --
~
D+, 0, 1) t,
(0, n_, 0, 1)t,
(1 + 7b)~/1 + n 2_
1 + 7b
and
~7---(0, (1 + 7b)~/1 + n 2
1
v/~(O, O, O, O, O, O, O, 1, O, O, O, 1, O, O, 1, O)t, 1
Vrj(1, 0, 0, 1,0, 1, O, O, O, O, O, O, O, O, O, O)t, 1
0,0,0,0,0,0,
bD++
1+--2 2 ' o,o,
X//X- (1 + D2+)
.~/-'f_-b2
l + b ) t, b(1 + D+), 0, 0, b(1 + D+), 0, bD+ + - -
V ---~'
d
1
l+b
b 0, 0, 0, 0, 0, 0, bD_ + - - ~ ,
0, 0,
V/X+(1 -4- D2_) ,~_ - b 2 l+b) V - - - 7 ' b(1 + D_), 0, 0, b(1 + D_), O, bD_ + - where
)v-l- ---D+ =
1+ b +
6 b 2 -4- ~/1
+ 2b +
b2
+ 20b 3 +
40b 4
1 + b - 2b 24- ~/1 + 2b + b 2 + 20b 3 + 40b 4 2b(1 + 3b)
t,
J
HERMITIAN TENSOR PRODUCTAPPROXIMATIONOF COMPLEXMATRICES
o,oo] (7 {0i 01)
Using (28) we have
b
Pb--2(1+7b)
/01"~
1010
®
1
b
-i'~
2(1+7b)
0
+
0101
)
®
(o+o) (l+7b)(l+D
2)
0
--Tl+b 0 b D+ ®
q (1 + 7 b ) ( 1 + D2_)
(o_o) 0
0
i
0 -i
0
b(1 + D+)
0
0
b(1 + D+)
0
0
172
1+b2+ bD+]
0
0
1-b 2 2
b(1 + D_)
0
0
0
b(1 + D_)
0
0
0
~-!bD_ ®
0 -i
0
1
~
1
-i
0
0010
q
283
1
172
l+b + 2
bD_
"~
I
(30)
4.
Separability of bipartite mixed states In this section we discuss some properties related to the Hermitian tensor product decomposition that could give 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 = ~ = 1 Bi ® Ci. Nevertheless, the Hermitian matrices Bi and Ci are generally not positive• They are not density matrices defined on the subspaces H1 and //2. Hence one cannot say that A is separable. Let re(A) and M(A) denote the smallest and the largest eigenvalues of a Hermitian matrix A. We can transform the decomposition into the one given by another set of Hermitian matrices which all have the smallest eigenvalue zero, as follows, £ r A = Bi ® Ci = E ( B i - m ( B i ) [ m + m(Bi)Im) ® (Ci - m ( C i ) I n + m ( C i ) I n ) i=l i=l r
r
= ~ . ( 8 i - m(Bi)Im) ® (G - m ( G ) I . ) + ~ m ( G ) ( ~ i - m(~OIm) ® I. i=l i=1 £ r +Im N m(Bi)(Ci - m(Ci)I~) + Em(Bi)m(Ci)Im ® In i=l i=l
S. M. FEi, N. JING and B. Z. SUN
284 r
= ~(Bi
- m(Bi)Im) ® (Ci
i=1
-- m(Ci)In)
I r
+ Zm(Ci)(Bi-m(Bi)Im)-m
i=1 +Im ®
( Zrm ( C i ) ( B i - m ( B i ) I m ) i=1
m(Bi)(Ci - m(Ci)In) - m
i=1
Zm(Bi)(Ci
) Im ] ®In
- m(Ci)In) In
i=1
r
q-[m(i~=l m(fi)ni) -~m(~m(ni)ci) -~m(Bi)m(fi)]lm i=1
(31)
where Im and In stand for an m x m and n x n identity matrices. The coefficient
Of
Im @In,
qB,C -= m
m(Ci)Bi
+m
m(Bi)Ci
-
i=1
m(Bi)m(Ci)
i=1
associated with the decomposition A = }-~=1 Bi @ Ci is not necessary positive. qe,c depends on decomposition. Associated with another decomposition A = Y~f-1 B; ® C~ one would obtain qB',c' 7~ qB,c. We define the maximum value of qB,C to be the separability indicator of A, S ( A ) = max(qs,c) for all possible Hermitian decompositions of A. With respect to S(A) the associated decomposition is generally of the form A = ~
i
[~i ® d i + Im® d + B ® In + S(A)Irn ® 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 the tensor product decompositions into Hermitian matrices like A = ~ = 1 Bi Q Ci. A is separable if and only if the separability indicator S(A) > O. Moreover, S(A) satisfies the following relations: S(A) < m ( a ) , (33)
1 S(A) > ~
1"
Z[M(Bi)m(Ci) + m(Ci)m(Bi) i=1
(34)
- I m ( B i ) l ( M ( C i ) - m(Ci)) - Im(Ci)l(M(Bi) - m(Bi))],
S(A) > re(A) - Z M([~i)M(Ci).
i
(35)
Proof: If A is separable, A has a decomposition A = ~J-~iBi Q Ci of the form (1), i.e. m(Bi) > O, m(Ci) > O. We have
285
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
S(A)>qB,c = m ( ~ m ( C i ) B i ) + m ( ~ i
m(Bi)Ci)-~, m(Bi)m(Ci)
>_ Zm(Ci)m(Bi)..q- ~ m ( B i ) m ( C i ) - Zm(Bi)m(Ci) i
i
i
= Zm(Bi)m(Ci) > O. i If S(A) > O, then the associated decomposition (32) is already a separable expression of A. From the decomposition (32) with respect to S(A) we have
m(A) > Z m(Bi Q Ci) + m([~) + m(C) + S(A) = S(A). i On the other hand we have
S(A) > qB,C >~Zm(m(Ci)Bi) + Zm(m(Bi)Ci) - Zm(Bi)m(Ci) i
i
i
= Z ( m(Bi)m(ci)+lm(Ci)l+ M(Bi)m(Ci)-Im(Ci)J) i 2 2 + Z ( m ( C i ) m(Bi)+lm(Bi)l +M(Ci)m(Bi)21m(Bi)l ) i 2 - Zm(Bi)m(Ci), i which is just the formula (34). By using the relations M(B + D) < M(B) + M(D), m(B + D) > m(B) m(B + D) < m(B) + M(D) for any m × m matrices B and D, we have
m(A) = m ( Z
+ m(D),
[~i ~ Ci "-~ Im ~ C -~- B ~ In q- S ( A ) I m @ In) i
< m(Im ® C + [~ ® In + S(Z)Im ® 11,)+ M( Z [~i ~ Ci) i
=m(B)+m(C)+ S ( A ) + M ( ~ .
Bi@Ci) l
< S(A) + i Hence formula (35) follows.
M([~i)M(Ci). []
r Generally, qB,C with respect to our decomposition A = Y~i=l Bi Q Ci is not equal to the separability indicator S(A). Suppose we have another decomposition A = Y~Tt=l n;~c~. As B i (and Ci) are defined in terms of the singular value decomposition eigenvectors, they are linearly independent. We can choose linear functionals q9i such that qgi(Bj) -~- ~ij" Applying qgi®l to both sides of Zi=lr ni@ci = ZTL1 Bti@Ci: we get
286
Ci
:
s . M . FEI, N. JING and B. Z. SUN F!
Zi=I
~°i (Bj)Cj, t i
i.e. Ci ~ (C 1! .
Cr, . ,, }. Similarly . .
we
. h a v e . Bi E
r!
r
(Brl . . . .
B r')" !
I
Therefore, any other Hermitian decomposition A = ~ i = 1 Bi ® Ci can be obtained r from our decomposition A = ~--~-i=1Bi ® Ci in terms of the following transformations:
Bj :
EijBi, i=l
Cj =
FijCi, i=1
as long as the real matrices E = (Eij) and F = (Fij) satisfy the relation E F t : Ir. The inequalities (33)-(35) can serve as the separability criterion themselves. For instance, if the minimum eigenvalue of A is zero, then A is entangled if the right hand side of (34) is great than zero. 5.
Conclusion and remarks
We have developed a method approximating general complex matrices by Hermitian tensor product. An explicit construction of density matrices on H1 ®/42 in terms of the sum of tensor products of Hermitian matrices on H1 and //2 is presented. Using 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 terms depends on the rank of A22), similar to the Schmidt decomposition for bipartite pure states. In example (30) we see that the 8 × 8 density matrix Pb has only 4 terms in the tensor product decomposition. In [27] an entanglement measure called robustness is introduced. For a mixed state p and a separable state Ps, the robustness of p relative to Ps, R(pllps), is the minimal s >_ 0 for which the density matrix (p + Sps)/(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 p is the one when Ps is taken to be the (separable) identity matrix. In this case p has the form p = (1 + t)p + - t I m ® In/mn), where p+ is separable, p is separable if and only if the minimum of t is zero. Therefore, the separability indicator appearing from our matrix decompositions is basically the minus of the random robustness, up to a normalization. Another interesting separable approximations of density matrices were presented in [28]. This method, the 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. Instead of 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. The results can be generalized to multipartite states. Let us consider a general /-partite mixed state Pl,2,...,l on a space HI ®/42 ® . . . ® Hi. We can first consider Pl,2,...,1 as a bipartite state on the space H1 and //2 ® - . . ®//1. Using Theorem 2 we can find the tensor decomposition Pl,Z,...,t : ~ 1 B/1 ® B/23l, where B/1 and B~ 3"''1
HERMITIAN TENSOR PRODUCT APPROXIMATION OF COMPLEX MATRICES
287
are Hermitian matrices on H1 and H2 ® . . . @ tit, respectively. The matrices B 23"''1 can be again decomposed as BZi3"•'1= y-~2=1 B~ ® B3ij"''t, ¥ i, with B/} and Bij3...t being Hermitian matrices on H2 and /-/3 @ . . . ® H1 respectively. Doing so we have the Hermitian tensor product decomposition of the form P1,2,...,l = }--~7=1 B] ®B/2®'" .@B~, where B/~ are Hermitian matrices on Hk. New decompositions can be obtained, Pl,e,...,1 . .Y-~;'-I B'~ ® B 'li' where B 'kj = Z i =r I E i jk B i k . .k . = . .1, . . . @ . B'2i @ l,
E k = (E/~) are the real matrices satisfying Y-~}'-I Elt l J .E2. t2J
• . .
E(.tl J = 6ili26i2i3
For any given decomposition, in terms of the protocol (31), one has
3it li~.
• .. _ Pl,2,...,1
=
B']• ® " ® . . . + B i + q Idl ® Id2 ® • • • ® Idt, where Idi is the identity matrix on Hi, (B'], B'~ . . . . . B'ti) are Hermitian matrices such that m(B'ki)= 0, or part of them (but not all) are identity matrices. The separability indicator S ( P l , 2 , . . . , l ) is the maximal value of the parameter q for all possible positive Hermitian tensor product decompositions. If the parameter S(pl,2,...,1) > 0, the state Pl,2,...,l is separable, otherwise it is entangled. Acknowledgements S. M. Fei gratefully acknowledges the warm hospitality of Dept. Math., NCSU and the support provided by American Mathematical Society. N. Jing greatly acknowledges the support from the NSA and Alexander von Humboldt foundation. REFERENCES [1] J. Preskill: The Theory of Quantum Information and Quantum C o m p u t a t i o n , California Inst. o f Tech., 2000, http://www.theory.caltech.edu/people/preskill/ph229/• [2] M.A. Nielsen and I.L. Chuang: Quantum Computation and Quantum I n f o r m a t i o n , Cambridge University Press, Cambridge 2000. [3] D. Bouwmeester, A. Ekert and A. Zeilinger (eds.): The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation and Quantum Computation, Springer, New York 2000. [4] M. Lewenstein, D. Bruss, J. I. Cirac, B. Kraus, M. Kug, J. Samsonowicz, A. Sanpera and R. Tarrach: J. Mod. Opt. 47 (2000), 2841. [5] B. M. Terhal: Theor. Comput. Sci. 287 (2002), 313. [6] M. Horodecki, R Horodecki and R. Horodecki: Springer Tracts in Mod. Phys. 173 (2001), 151. [7] J. S. Bell: Physics (N.Y.) 1 (1964), 195. [8] A. Peres: Phys. Rev. Lett. 77 (1996), 1413. [91 M. Horodecki, E Horodecki and R. Horodecki: Phys. Lett. A 223 (1996), 1. [10] M. Horodecki and R Horodecki: Phys. Rev. A 59 (1999), 4206. [11] N. J. Cerf, C. Adami and R. M. Gingrich: Phys. Rev. A 60 (1999), 898. [12] M. A. Nielsen and J. Kempe: Phys. Rev. Lett. 86 (2001), 5184. [13] B. Terhal: Phys. Lett. A 271 (2000), 319. [14] M. Lewenstein, B. Kraus, J. I. Cirac and R Horodecki: Phys. Rev. A 62 (2000), 052310. [15] O. Rudolph: Physical Review A 67 (2003), 032312. [16] K. Chen and L. A. Wu: Quant. Inf. Comput. 3 (2003), 193. [17] K. Chen and L. A. Wu: Phys. Lett. A 306 (2002), 14. [18] S. Albeverio, K. Chen and S. M. Fei: Phys. Rev. A 68 (2003), 062313. [19] R Horodecki, M. Lewenstein, G. Vidal and I. Cirac: Phys. Rev. A 62 (2000), 032310. [20] S. Albeverio, S. M. Fei and D. Goswami: Phys. Lett. A 286 (2001), 91. [21] S. M. Fei, X. H. Gao, X. H. Wang, Z. X. Wang and K. Wu: Phys. Lett. A 300 (2002), 555.
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