- Email: [email protected]
Separability criterion and inseparable mixed states with positive partial transposition
4 August
1997
PHYSICS
LETTERS
A
Physics Letters A 232 ( 1997) 333-339
ELSEVIER
Separability criterion and inseparable mixed states with positive partial transposition Pawei Horodecki Faculty of Applied Physics and Mathematics, Technical University of Gdarisk, 80-952 Gdarisk, Poland Received 26
March 1997; revised manuscript received 5 June 1997; accepted for publication 5 June 1997 Communicatedby V.M.Agranovich
Abstract
It is shown that any separable state on the Hilbert space 7-1= XI 8 l-t2 can be written as a convex combination of N pure product states with N < (dim’H)‘. Then a new separability criterion for mixed states in terms of the range of the density matrix is obtained. It is used in the construction of inseparable mixed states with positive partial transposition in the case of 3 x 3 and 2 x 4 systems. The states represent an entanglement which is hidden in a more subtle way than known so far. @ 1997 Published by Elsevier Science B.V.
1. Introduction The problem of quantum inseparability of mixed states has attracted much attention recently and it has been widely considered in different physical contexts (see Ref. [ 1] and references therein). In particular an effective criterion of separability of 2 x 2 and 2 x 3 systems has been obtained [ 2,3]. Quite recently this criterion has been used for characterisation of the twobit quantum gate [ 41 and quantum broadcasting [ 51. It also enabled one to show that any inseparable state of the 2 x 2 system can be distilled to a singlet form [ 61. We recall that the state e acting on the Hilbert space ti = El @ 3-12is called separable if it can be written or approximated (in the trace norm) by states of the form
Usually one deals with the case dim7-t = m. For this case it will appear subsequently that any separable state can be written as a convex combination of finite product pure states, i.e., in those cases the “approximation” part of the definition appears to be redundant. Peres has shown [2] that the necessary condition for separability of the state Q is the positivity of its partial transposition eTz. The latter associated with an arbitrary product orthonormal fi @ fjbasis is defined by the matrix elements, e ZP,ny = (fm @ _f,leTzl_fn8
fvj = emv,np.
(2)
Although the matrix eTz depends on the used basis, its eigenvalues do not. Consequently, for any state the condition can be checked using an arbitrary product orthonormal basis ’ . It has been shown [ 31 that for 2 x
k @ =
C
Pi&
@
Oi9
(1)
i=l
where pi and @iare states on LX1and ‘Hz, respectively. 0375.96() I,/97/S 17.00 0 I997 Elsevier Science B.V. All rights reserved. P/I S0?75~9601~97~00416-7
’ As the full transposition of a positive operator is also positive, the positivity of the partial transposition ~~2 is equivalent to the positivity of the (defined in an analogous way) partial transposition eTl
P. Horodecki/Pilyslc.~
334
2 and 2 x 3 systems the partial transposition condition is also a sufficient one. Thus, in those cases the set of separable states has been characterized completely in a simple way. For higher dimensions the necessary and sufficient condition for separability has been provided [3] in terms of positive maps. Namely, the state e acting on the Hilbert space ‘FI = tit @ ‘Hz is separable iff for any positive map A : B(‘Hx) -+ L?(%I> the operator I @ A@ is positive (a(‘&) denote the set of all operators acting on ‘Fli and I is the identity map). Then a natural question arose, whether the partial transposition condition is also sufficient for higher dimensions. The negative answer to this question has been established [ 31, however, with no explicit counterexample being given. In this Letter we provide (Section 3) a new criterion for the separability of quantum states. It is done with the help of the analysis of the range of density matrices via the decomposition of separable states on pure product states (Section 2). Namely, it appears that any separable states can be written as a convex combination of a finite number N of pure product states with N restricted by the squared dimension of the respective Hilbert space. In Section 4 we construct families of inseparable states with positive partial transposition for 3 x 3 and 2 x 4 systems. We achieve our goal using the separability criterion and the technique introduced by Woronowicz in Ref. [ 91 which has provided a heuristic basis for the present analysis. It appears that, in general, the new criterion is rather independent of than equivalent to the partial transposition one.
2. Finite decomposition
of separable states
c
Pik p+, @
Qa.
Msep = conv Pep.
(4)
Here conv A denotes the convex hull of A and means the set of all possible finite convex combinations of elements from A 2 . B stands for the closure of B in the trace norm topology. It appears as a standard fact from the convex set theory that the convex hull of any compact set from a finite-dimensional space is compact itself (see Ref. [7], Theorem 14, p. 210). Thus the closure of M, = conv Psep = conv Psp. Hence the set of extremal points of M,, is equivalent to Psep. Then we can apply the Caratheodoty theorem [S] which says that any element of a compact convex subset of R” (in our case n = m2 - 1) can be represented as a convex combination of (at most n + 1) finely independent extreme points [ 81. Usage of this theorem completes the proof of our statement.
First we will need the following.
Theorem 1. Let e be a separable state acting on the Hilbert space X = 3-11@ 7i2, dimIH = m < CO. Then there exists a set of N, N < m2 product projectors P$, @IQ~,, {i, k} E I (I is a finite set of pairs of indices with number of pairs N = #I < m2) and probabilities Pik such that
{i.k}EI
ProojI The proof depends on the properties of compact convex sets in real finite dimensional spaces. Namely, then the set of separable states M,, defined above can be treated as a compact convex subset of finite-dimensional real space (obtained by real linear combinations of Hermitian operators) with a dimension n = m2 - 1. The set of separable states M,,, is a closed convex subset of the convex set of all quantum states P which is bounded, compact and given by n = m2 - 1 real parameters in the Hilbert-Schmidt basis. Let us denote by Psep c Msep the set of all separable pure states. Psep is obviously compact (it is a tensor product of two spheres which are compact in the finitedimensional case). We have from the definition of Msep immediately,
3. Separability criterion
We prove here the following.
e =
Lrtters A 232 (I9971 333-339
(3)
Lemma X, dim8
1. Let state e act on the Hilbert space < 00. Then for an arbitrary @ensemble
{pi,Pi},
2 The convex hull of the set of A is usually defined as a minimal convex set containing A, but it is shown that it is equivalent to the set of all possible finite convex combinations of an element of A (see Ref. [71).
P. Horodecki
/ Physics
(5)
Lerrers A 232 I 1997) 333-339
@”
pikp$;
=
335
8
Qf,
i=l
each of the vectors qi belongs to the range of the state @ Proo$ The range of e is defined by Ran e E {fi E ‘H: ~4 = @ for some Cp E 3-1). As @ is a linear and Hermitian operator we have that Ran Q is simply a subspace spanned by all eigenvectors of @ belonging to nonzero eigenvalues. In short, Ran e is a support of e. Following Ref. [ lo] we have that ‘Pi belongs to the support of Q. Thus any !Pi belongs to Ran e. Now we can prove our main result. Theorem 2. Let e act on the Hilbert space E = Et 63 7-i~, dim’H = m. If Q is separable then there exists a Set of product VeCtOrS {(hi @3 &}, {i, k} E 1 (I is a finite set of pairs of indices with number of pairs N = #I < m2) and probabilities pik such that (i) the ensemble {$i @ &, pik} ({@i @ &, pik}) corresponds to the matrix Q ( eT2), (ii) the vectors {$i 63 +k} ({@i @ 4;)) span the range of e ( eTz ), in particular any of the vectors {(cli@ +k) ({@i @ 4;)) belongs to the range of e (@‘I). Proof Let us prove first statement (i). According to the theorem of Section 2 any separable state Q can be written in the form
Q =
C
pik pj,
@
hence we obtain the statement (i). Obviously, any vector r/l from the range of the state is given by a linear combination of vectors belonging to the ensemble realising the state. Using the lemma immediately completes the proof of (ii). Remark 1. Using the full transposition one can easily see that the analogous theorem (with vectors conjugated on the first space) equivalent to the above one is valid for $I. Remark 2. The conjugation @ associated with the basis the transposition of Q+ was performed in, is simply obtained by the complex conjugation of the coefficients in this basis up to the irrelevant phase factor. Then the operation of partial complex conjugation (we will denote it by P*2) can be illustrated as follows,
((w
+ Pe2> 8 (yeI + ae2))*2
=
(cuei +
pe2)
63 (ye1 + Se;?)*
=
(ael +
@2>
C3(y’el
S*e2),
(9)
where the standard basis et, e2 in C2 was used in the transposition of the corresponding projector. Note that the operation of partial conjugation is defined only for product vectors.
4. Inseparable transposition
Qcp,
+
states with positive partial
{i.k}EI E
c
pikI$i
@ 4k)($i
(6)
@ d’kl.
4. I. 3 x 3 system
ji,k}EI
Consider the Hilbert space 3-1= C3 8 C3. Let P# E
using only N = #I < m2 pure product states Pei @Q$, . Remembering that the transposition of the Hermitian operator is simply equivalent to the complex conjugation of its matrix elements we get
Q;, = Q;, =
(l+k)b%‘kl)*
=
14;)(4;1 = Q4;.
From the above and the definition sition (2) we obtain
(7)
of partial transpo-
/4)(41 and let {ei}, i = 1, 2, 3, stand for standard basis in C3. Then we define the projector
and vectors y-l(el~el+e2~e2+e3~e3),
6
(11)
P. Horodecki/
336
O
Physks
(12)
Now we define the following
state,
8a
+
1
-he. 8a+
and then check that the remaining operator in the combination is positive. Thus ~2 is a legitimate state. Now we will show that it is inseparable. Then, as the operation of partial transposition preserves separability, we will have two “dual” sets of inseparable mixtures with positive partial transposition. Let us find all product (unnormalised for convenience) vectors belonging to the range of ~3. We will adopt here the horizontal notation with the basis ordered as et @ et, et @ e2, ei @ es, e2 @ el, e2 @ e2, . . . and so on. Assume, in addition, that a # 0, 1. Then any vector belonging to the range of ~2 can be presented as
A,B,C,D,E,F
EC,
(16)
with nonzero x = J( 1 + a) ( 1 - a). On the other hand, if u is to be positive, it must be of the form 1
aOOOaO OaOOOO
OOaOOO OOOaOO
aOOOaO OOOOOa 000000 000000
0 0 0 0 0 0 $(l+a) 0
0 0 0 0 0 0
Uprod = (r,s,f)
tf 0 0 t
o;J1_az a
a000aO~J1-a20
0 ;('+a)
1
e,T2 --- 8a + 1 a00000 OaOaOO OOaOOO X
1
u=(A,B,C;B,D,E;C+EE,xF),
1
X
(15)
(14)
1
Its matrix and the matrix of its partial transposition are of the form
@‘= 8a+
133-3_?9
001 010 ( 100
(13)
This state is inseparable as its partial transposition possesses a negative eigenvalue A = ( 1 - &)/2 belonging to the eigenvector [2/(5 + fi>] (ei @ es + [ (-1 - &)/2]es @ et). Here inseparability comes from the highly entangled pure state Py. On the other hand, the state PO, corresponding to the vector ( 12) is evidently separable. Below we will see that it is possible to mix the states einq and PQ. in such a way that the resulting state will have partial transposition positive being nevertheless inseparable. For this purpose consider the following state,
f 1’197)
It is easy to show that ~2 is positive. Indeed it suffices only to single out the state I @ UPQ~I @ U’f as a component of a convex combination where
u=
@jinsep= ijPp + $.
@a = 8a + 1 @imp
Letter., A 232
OaOaOO OOOOaO OOOOOa OOaOOO OOOOOa
0 0 : 0 0 #+a)
0
oooooo~~i3o
0 0 0 0
0 a
0 0 0 0 0 0
o;&-z a
0 $+a)
z
8
(&&c)
(r(A,B,C);s(A,B,e);t(~,~,~)),
r,s,t,fi,B,d
EC.
(1 7)
Let us now consider the following cases: (i) rs # 0, then without loss of generality we can put r = 1 and A = A, fi = B, c = C. Comparison with (16) gives us in turn: B = sA; E = SC, E = tsA + C = tA (hence A # 0 or Uprodvanishes) ; xF = tC = t2A, C + F = tA with C = tA + F = 0; xF = tC = t2A in the presence of vanishing F and nonvanishing A + t = 0. Thus we obtain the states ~1 =A(l,s,0)@(1,s,O),
A,scC.
(18)
(ii) r = 0. Then we have Uprod= (0,0,O;s(A,~,~);t(A,B,e)), s,t,A,B,C
E c.
On the other hand, one gets
(19)
P. Horodecki/
Uprod
=
(O,O,O;
0,
D,
E;
F, E,
Phy.y.,ic.sLerrer.\ A 232 (1997) 33.3-339
xF),
D,E, F E C.
(20)
Now either s = 0 and then, according gives us
to (20) E = 0
F EC
u2 = F(O,O, 1) 8 (1,0,x),
(21)
or s # 0. In the last case we can put s = 1. Consequently it is possible that t = 0 and then we get via the conditions F = 0, E = 0 another product state, D EC.
u3 = D(0, l,O> @ (O,l,O>,
(A,O,C;O,O,O;t(A,O,C))
+F,O,xF).
(23)
Then for I = 0 we get C = F = 0 and ~4=A(1,0,0)@‘1,0,0),
AEC,
(24)
or, provided that t # 0, xF = tC, C + F = tA + A = (t-’ + x-‘)C and then ug =C(l,O,t) C,t E c, All partial (21),
(22),
@(~-‘+x-‘,o,l), I f 0. complex (24),
(25) conjugations
of vectors
A,sEC,
~;~=F(O,O,1)@(1,0,x),
FEC,
*2=D(0,1,0)@(0,1,0), 1’3
DEC,
ll;?
A
A(l,O,O)
U;2=C(l,o,t) c, t E c,
c3
(18),
(28) where projectors Po(@, correspond to the vectors a($) = (1/&)(1,ei@,e-2i~). Note that the integral representation (28) is not unique. The representation of the “dual” state ~2 is obtained by complex conjugation of projectors acting on the second space. For the case a = 0 we get simply the product state PO, (cf. ( 12) ) . Taking the parameter a arbitrarily close to 0, we obtain almost product pure states PQ, being nevertheless separable. The situation is, in a sense, analogical to the case of the states introduced in Ref. [ 121. The inseparability of the latter was also determined by a parametric change of both coherences and probabilities involved in the state. 4.2. 2 x 4 system
(25) are
~;~=A(l,s,O)@(l,s*,O),
=
which belongs just to Ran ea. Hence, for any c1 # 0, 1, the state c7;fi violates the condition due to the second statement of the theorem. Thus the state is inseparable together with the “dual” original state ea. In the latter the state PO, masks the inseparability due to einxp, making it “invisible” to the partial transposition criterion, but does not destroy it. It is interesting to see the limit behaviour of the state ea. In the case of a = 0 we get the separable state with the symmetric representation
(22)
For the case t # 0 we get from ( 19)) (20) j = 0 + F = 0 =+ E = 0 + D = 0. Hence the only product vector with nonvanishing z is the trivial zero vector. (iii) r # 0, s = 0. As in the case of (i) we can put r = 1 and A = A, B = B, c = C. Then we have B = E = D = 0 which leads to the equality
= (A,O.C;O,O,O;C
337
(1,&O),
@((P-l t # 0.
E
s # 0,
Here we will use the vectors i = 1,2,3,
(29)
C,
+x-‘,O,l),
O
(30)
(26) Then we can construct the following
state,
It is easy to see that the above vectors cannot span the range of @a as they are orthogonal to the vector (31) ti = (O,O, 1) @ (O,l,O).
(27)
i=l
which is inseparable (it can be easily verified like in the state ginxp using a partial transposition criterion). Now the states of our interest are of the form 7b cb
=
The corresponding
(32)
-by
7bfl
matrices are 3
bOO0
0
b0
0
ObOO OObO
0
Ob
0
0
00
b
OOOb
0
00
0
’ I 0000 bOO0 ObOO
;(l+b) 0 0
00&/s 60 Ob
0 0
OObO;J1_bZOO
u;*=F(O,l)@(l,O,O,y),
FEC,
u~*=D(1,0)~(0,0,0,1),
DEC.
bOO0 ObOO OObO OOOb
0 b 0 0
Let us check now whether the vectors (37) can be written in the above form. For the u;*, assuming that it is a nontrivial one (C f 0) and at the same time is of the form (38)) taking into account the coefficient Cl we obtain
;(l+b)
ObOO OObO OOOb
$(l+b) 0 0
00 00 60 Ob
0 0 0 0
OO;J1-b2 b0 0 Ob 0 ;(l+b)
I
(33)
It is easy to see that the state u:’ is positive as (34)
l\
J .
(37)
(38)
s*=s -1 .
0000~~~00
“=
(36)
A’, B’,C’, D’, E’ E C.
76-k 1
0010 0100 i 1000
s # 0,
w=(A’,B’,C’,D’;B’+yE’,C’,D’,E’),
1
/ooo
c, s E c,
On the other hand, any vectors from the range of a;f2 can be written in our notation as
(Tb=7b+1
X
of all possible product vectors ui E
1 +
~6nsep
clbT2 _--
plex conjugations Ran fib,
(35)
Consequently, for y = J(1 - b)/( 1 + b) f 0,l we get, analogically as Section 4.1, the partial com-
(39)
On the other hand, considering we have s* + ys(s*)-‘(1
+ y(s*>s)
B’, E’ and B’ + YE’,
= s(s*)?
(40)
Finally taking into account D’ we obtain (SW)-‘(l+y(s*)3)
=s.
(41)
Combining all the three equations above we find that ys* = 0, s Z 0 which contradicts the fact that y does not vanish. For uG2we obtain yE’ = D and at the same time E’ = yD, which is impossible for y # 0,l unless E’ = D = 0 trivialising then the vector u;~. For the vector u;* we get immediately that D = 0 must hold. It leads to the conclusion that none of vectors ur2 belongs to the Ran ~2 apart from the trivial zero one. Thus for any b # 0, 1 the state ffb violates our criterion from the theorem (statement (ii) ) being then inseparable together with its “dual” counterpart ~2. Here again the limit cases correspond to separable states. Namely we have go=s “PP , + $(Pc,c3e4+
pe,c4e,
1
.1The example of the pair of matrices of such a type treated, however, as operators on C4 63 C4 together with a similar analysis of their ranges has been considered in Ref. 191 in the context of positive maps.
(42)
P. Horodecki/
P/yic-.\ I.er~er.\A 232 ( 1997) .<33-339
where $(4) = (I/fi)(l,e’4) and F(4) = ( l/2) ( 1, eei6, ew2@, e-3i6). Putting b = 1 we obtain again the separable state Pp, . Thus we have provided the families of inseparable states with positive partial transposition. It is natural to ask how they are related to the necessary and sufficient separability condition given in terms of positive maps (Ref. [ 31, see Introduction). Clearly it follows that, in the presence of inseparability of states &, Ub, there must exist positive maps A, : L?(C3) ---f a(C3) and L&, : B(C4) + B(C2) such that the operators I @ Aa& and I @ &ffb are not positive, i.e., each of them possesses at least one negative eigenvalue. It is easy to see that the maps A,, Ah cannot be of the form A=Ay+AyT,
339
transposition criterion and they would also satisfy the present one as the latter is useful only for the states with a range essentially less than 7-L Now, as the set of separable states is closed, taking sufficiently small E one can ensure that the new states remain inseparable, despite their satisfying both criteria. The present criterion allowed us to provide examples of states of a new kind, where the entanglement is masked in a specific way by a classical admixture. In this context the interesting problem arises whether it is possible to distill such an entanglement using local operations and classical communication.
Acknowledgements
(43)
where &’ are completely positive maps and T is a transposition [ 31. However, the nature of the positive maps which are not of the form (43) is not known yet and finding the maps A,, Ab revealing the inseparability of the states pa, gb may be difficult.
5. Conclusion We have pointed out that any separable state can be written as a convex combination of only N pure product states ( N < (dim 31) 2, _We have provided a new necessary condition for separability of quantum states in terms of the range of density matrices. For any separable state it must be possible to span its range by a system of such product vectors that their counterparts obtained by partial complex conjugation span the range of partial transposition of the state. It is interesting to see that the above criterion sometimes does not reveal inseparability in cases where the partial transposition one works (this can be seen for the case of 2 x 2 states [ Ill) but it happens to be efficient where the latter fails. Thus, both the criteria are, in general, independent for mixed states, although one can easily verify (via Schmidt decomposition) their equivalence for pure states. One could suppose that, taken jointly, they can constitute the new necessary and sufficient condition of separability in higher dimensions. However, it is not the case [ 131: one can take the states a # 0,1, ((I-~)0-b+d/4,b # (1 -E)e,+E1/9, 0, 1). Those states will obviously satisfy the partial
The author is indebted to A. Sanpera for drawing his attention to the problem and for useful discussions. He also thanks T. Figiel and J. Popko for consultations on convex sets theory and R. Alicki, R. Horodecki and M. Horodecki for helpful comments. Special thanks are due to A. Peres for remarks leading to a significant improvement of the letter. This work is supported in part by the Polish Committee for Scientific Research, Contract No. 2 P03B 024 12. References [ I] C.H.
Bennett, D.P. Di Vincenzo, J. Smolin and W.K. Wootters, Phys. Rev. A 54 (1997) 3814. ]2] A. Penes, Phys. Rev. Lett. 76 (1997) 1413. [3] M. Horodecki, P Horodecki and R. Horodecki, Phys. Len. A 223 (1996) 1. [4] J.E Poyatos, J.I. Cirac and P Zoller, Phys. Rev. Lett. 78 (1997) 390. [ 51 V. Buick, V. Vedral, M.B. Plenio, PL. Knight and M. Hillery, Phys Rev. A 55 (1997) 3327. [6] M. Horodecki, P Horodecki and R. Horodecki, Phys. Rev. L&t. 78 (1997) 574. [ 71 PJ. Kelly and M.L. Weiss, Geometry and Convexity (John Wiley, New York, 1979). [ 81 E.M. Alfsen, Compact Convex Sets and Boundary Integrals ( Springer-Verlag, Berlin, 197 1) [9] S.L. Woronowicz, Rep. Math. Phys. 10 ( 1976) 165. [ IO] L.P Hughston, R. Jozsa and W.W. Wooters, Phys. Len. A 183 (1993) 14. [ 1I] R.E Werner, Phys. Rev. A 40 (1989) 4277. [ 121 R. Horodecki, Phys. Lett. A 210 (1996) 223. [ 131 A. Peres, private communication.
1997
PHYSICS
LETTERS
A
Physics Letters A 232 ( 1997) 333-339
ELSEVIER
Separability criterion and inseparable mixed states with positive partial transposition Pawei Horodecki Faculty of Applied Physics and Mathematics, Technical University of Gdarisk, 80-952 Gdarisk, Poland Received 26
March 1997; revised manuscript received 5 June 1997; accepted for publication 5 June 1997 Communicatedby V.M.Agranovich
Abstract
It is shown that any separable state on the Hilbert space 7-1= XI 8 l-t2 can be written as a convex combination of N pure product states with N < (dim’H)‘. Then a new separability criterion for mixed states in terms of the range of the density matrix is obtained. It is used in the construction of inseparable mixed states with positive partial transposition in the case of 3 x 3 and 2 x 4 systems. The states represent an entanglement which is hidden in a more subtle way than known so far. @ 1997 Published by Elsevier Science B.V.
1. Introduction The problem of quantum inseparability of mixed states has attracted much attention recently and it has been widely considered in different physical contexts (see Ref. [ 1] and references therein). In particular an effective criterion of separability of 2 x 2 and 2 x 3 systems has been obtained [ 2,3]. Quite recently this criterion has been used for characterisation of the twobit quantum gate [ 41 and quantum broadcasting [ 51. It also enabled one to show that any inseparable state of the 2 x 2 system can be distilled to a singlet form [ 61. We recall that the state e acting on the Hilbert space ti = El @ 3-12is called separable if it can be written or approximated (in the trace norm) by states of the form
Usually one deals with the case dim7-t = m. For this case it will appear subsequently that any separable state can be written as a convex combination of finite product pure states, i.e., in those cases the “approximation” part of the definition appears to be redundant. Peres has shown [2] that the necessary condition for separability of the state Q is the positivity of its partial transposition eTz. The latter associated with an arbitrary product orthonormal fi @ fjbasis is defined by the matrix elements, e ZP,ny = (fm @ _f,leTzl_fn8
fvj = emv,np.
(2)
Although the matrix eTz depends on the used basis, its eigenvalues do not. Consequently, for any state the condition can be checked using an arbitrary product orthonormal basis ’ . It has been shown [ 31 that for 2 x
k @ =
C
Pi&
@
Oi9
(1)
i=l
where pi and @iare states on LX1and ‘Hz, respectively. 0375.96() I,/97/S 17.00 0 I997 Elsevier Science B.V. All rights reserved. P/I S0?75~9601~97~00416-7
’ As the full transposition of a positive operator is also positive, the positivity of the partial transposition ~~2 is equivalent to the positivity of the (defined in an analogous way) partial transposition eTl
P. Horodecki/Pilyslc.~
334
2 and 2 x 3 systems the partial transposition condition is also a sufficient one. Thus, in those cases the set of separable states has been characterized completely in a simple way. For higher dimensions the necessary and sufficient condition for separability has been provided [3] in terms of positive maps. Namely, the state e acting on the Hilbert space ‘FI = tit @ ‘Hz is separable iff for any positive map A : B(‘Hx) -+ L?(%I> the operator I @ A@ is positive (a(‘&) denote the set of all operators acting on ‘Fli and I is the identity map). Then a natural question arose, whether the partial transposition condition is also sufficient for higher dimensions. The negative answer to this question has been established [ 31, however, with no explicit counterexample being given. In this Letter we provide (Section 3) a new criterion for the separability of quantum states. It is done with the help of the analysis of the range of density matrices via the decomposition of separable states on pure product states (Section 2). Namely, it appears that any separable states can be written as a convex combination of a finite number N of pure product states with N restricted by the squared dimension of the respective Hilbert space. In Section 4 we construct families of inseparable states with positive partial transposition for 3 x 3 and 2 x 4 systems. We achieve our goal using the separability criterion and the technique introduced by Woronowicz in Ref. [ 91 which has provided a heuristic basis for the present analysis. It appears that, in general, the new criterion is rather independent of than equivalent to the partial transposition one.
2. Finite decomposition
of separable states
c
Pik p+, @
Qa.
Msep = conv Pep.
(4)
Here conv A denotes the convex hull of A and means the set of all possible finite convex combinations of elements from A 2 . B stands for the closure of B in the trace norm topology. It appears as a standard fact from the convex set theory that the convex hull of any compact set from a finite-dimensional space is compact itself (see Ref. [7], Theorem 14, p. 210). Thus the closure of M, = conv Psep = conv Psp. Hence the set of extremal points of M,, is equivalent to Psep. Then we can apply the Caratheodoty theorem [S] which says that any element of a compact convex subset of R” (in our case n = m2 - 1) can be represented as a convex combination of (at most n + 1) finely independent extreme points [ 81. Usage of this theorem completes the proof of our statement.
First we will need the following.
Theorem 1. Let e be a separable state acting on the Hilbert space X = 3-11@ 7i2, dimIH = m < CO. Then there exists a set of N, N < m2 product projectors P$, @IQ~,, {i, k} E I (I is a finite set of pairs of indices with number of pairs N = #I < m2) and probabilities Pik such that
{i.k}EI
ProojI The proof depends on the properties of compact convex sets in real finite dimensional spaces. Namely, then the set of separable states M,, defined above can be treated as a compact convex subset of finite-dimensional real space (obtained by real linear combinations of Hermitian operators) with a dimension n = m2 - 1. The set of separable states M,,, is a closed convex subset of the convex set of all quantum states P which is bounded, compact and given by n = m2 - 1 real parameters in the Hilbert-Schmidt basis. Let us denote by Psep c Msep the set of all separable pure states. Psep is obviously compact (it is a tensor product of two spheres which are compact in the finitedimensional case). We have from the definition of Msep immediately,
3. Separability criterion
We prove here the following.
e =
Lrtters A 232 (I9971 333-339
(3)
Lemma X, dim8
1. Let state e act on the Hilbert space < 00. Then for an arbitrary @ensemble
{pi,Pi},
2 The convex hull of the set of A is usually defined as a minimal convex set containing A, but it is shown that it is equivalent to the set of all possible finite convex combinations of an element of A (see Ref. [71).
P. Horodecki
/ Physics
(5)
Lerrers A 232 I 1997) 333-339
@”
pikp$;
=
335
8
Qf,
i=l
each of the vectors qi belongs to the range of the state @ Proo$ The range of e is defined by Ran e E {fi E ‘H: ~4 = @ for some Cp E 3-1). As @ is a linear and Hermitian operator we have that Ran Q is simply a subspace spanned by all eigenvectors of @ belonging to nonzero eigenvalues. In short, Ran e is a support of e. Following Ref. [ lo] we have that ‘Pi belongs to the support of Q. Thus any !Pi belongs to Ran e. Now we can prove our main result. Theorem 2. Let e act on the Hilbert space E = Et 63 7-i~, dim’H = m. If Q is separable then there exists a Set of product VeCtOrS {(hi @3 &}, {i, k} E 1 (I is a finite set of pairs of indices with number of pairs N = #I < m2) and probabilities pik such that (i) the ensemble {$i @ &, pik} ({@i @ &, pik}) corresponds to the matrix Q ( eT2), (ii) the vectors {$i 63 +k} ({@i @ 4;)) span the range of e ( eTz ), in particular any of the vectors {(cli@ +k) ({@i @ 4;)) belongs to the range of e (@‘I). Proof Let us prove first statement (i). According to the theorem of Section 2 any separable state Q can be written in the form
Q =
C
pik pj,
@
hence we obtain the statement (i). Obviously, any vector r/l from the range of the state is given by a linear combination of vectors belonging to the ensemble realising the state. Using the lemma immediately completes the proof of (ii). Remark 1. Using the full transposition one can easily see that the analogous theorem (with vectors conjugated on the first space) equivalent to the above one is valid for $I. Remark 2. The conjugation @ associated with the basis the transposition of Q+ was performed in, is simply obtained by the complex conjugation of the coefficients in this basis up to the irrelevant phase factor. Then the operation of partial complex conjugation (we will denote it by P*2) can be illustrated as follows,
((w
+ Pe2> 8 (yeI + ae2))*2
=
(cuei +
pe2)
63 (ye1 + Se;?)*
=
(ael +
@2>
C3(y’el
S*e2),
(9)
where the standard basis et, e2 in C2 was used in the transposition of the corresponding projector. Note that the operation of partial conjugation is defined only for product vectors.
4. Inseparable transposition
Qcp,
+
states with positive partial
{i.k}EI E
c
pikI$i
@ 4k)($i
(6)
@ d’kl.
4. I. 3 x 3 system
ji,k}EI
Consider the Hilbert space 3-1= C3 8 C3. Let P# E
using only N = #I < m2 pure product states Pei @Q$, . Remembering that the transposition of the Hermitian operator is simply equivalent to the complex conjugation of its matrix elements we get
Q;, = Q;, =
(l+k)b%‘kl)*
=
14;)(4;1 = Q4;.
From the above and the definition sition (2) we obtain
(7)
of partial transpo-
/4)(41 and let {ei}, i = 1, 2, 3, stand for standard basis in C3. Then we define the projector
and vectors y-l(el~el+e2~e2+e3~e3),
6
(11)
P. Horodecki/
336
O
Physks
(12)
Now we define the following
state,
8a
+
1
-he. 8a+
and then check that the remaining operator in the combination is positive. Thus ~2 is a legitimate state. Now we will show that it is inseparable. Then, as the operation of partial transposition preserves separability, we will have two “dual” sets of inseparable mixtures with positive partial transposition. Let us find all product (unnormalised for convenience) vectors belonging to the range of ~3. We will adopt here the horizontal notation with the basis ordered as et @ et, et @ e2, ei @ es, e2 @ el, e2 @ e2, . . . and so on. Assume, in addition, that a # 0, 1. Then any vector belonging to the range of ~2 can be presented as
A,B,C,D,E,F
EC,
(16)
with nonzero x = J( 1 + a) ( 1 - a). On the other hand, if u is to be positive, it must be of the form 1
aOOOaO OaOOOO
OOaOOO OOOaOO
aOOOaO OOOOOa 000000 000000
0 0 0 0 0 0 $(l+a) 0
0 0 0 0 0 0
Uprod = (r,s,f)
tf 0 0 t
o;J1_az a
a000aO~J1-a20
0 ;('+a)
1
e,T2 --- 8a + 1 a00000 OaOaOO OOaOOO X
1
u=(A,B,C;B,D,E;C+EE,xF),
1
X
(15)
(14)
1
Its matrix and the matrix of its partial transposition are of the form
@‘= 8a+
133-3_?9
001 010 ( 100
(13)
This state is inseparable as its partial transposition possesses a negative eigenvalue A = ( 1 - &)/2 belonging to the eigenvector [2/(5 + fi>] (ei @ es + [ (-1 - &)/2]es @ et). Here inseparability comes from the highly entangled pure state Py. On the other hand, the state PO, corresponding to the vector ( 12) is evidently separable. Below we will see that it is possible to mix the states einq and PQ. in such a way that the resulting state will have partial transposition positive being nevertheless inseparable. For this purpose consider the following state,
f 1’197)
It is easy to show that ~2 is positive. Indeed it suffices only to single out the state I @ UPQ~I @ U’f as a component of a convex combination where
u=
@jinsep= ijPp + $.
@a = 8a + 1 @imp
Letter., A 232
OaOaOO OOOOaO OOOOOa OOaOOO OOOOOa
0 0 : 0 0 #+a)
0
oooooo~~i3o
0 0 0 0
0 a
0 0 0 0 0 0
o;&-z a
0 $+a)
z
8
(&&c)
(r(A,B,C);s(A,B,e);t(~,~,~)),
r,s,t,fi,B,d
EC.
(1 7)
Let us now consider the following cases: (i) rs # 0, then without loss of generality we can put r = 1 and A = A, fi = B, c = C. Comparison with (16) gives us in turn: B = sA; E = SC, E = tsA + C = tA (hence A # 0 or Uprodvanishes) ; xF = tC = t2A, C + F = tA with C = tA + F = 0; xF = tC = t2A in the presence of vanishing F and nonvanishing A + t = 0. Thus we obtain the states ~1 =A(l,s,0)@(1,s,O),
A,scC.
(18)
(ii) r = 0. Then we have Uprod= (0,0,O;s(A,~,~);t(A,B,e)), s,t,A,B,C
E c.
On the other hand, one gets
(19)
P. Horodecki/
Uprod
=
(O,O,O;
0,
D,
E;
F, E,
Phy.y.,ic.sLerrer.\ A 232 (1997) 33.3-339
xF),
D,E, F E C.
(20)
Now either s = 0 and then, according gives us
to (20) E = 0
F EC
u2 = F(O,O, 1) 8 (1,0,x),
(21)
or s # 0. In the last case we can put s = 1. Consequently it is possible that t = 0 and then we get via the conditions F = 0, E = 0 another product state, D EC.
u3 = D(0, l,O> @ (O,l,O>,
(A,O,C;O,O,O;t(A,O,C))
+F,O,xF).
(23)
Then for I = 0 we get C = F = 0 and ~4=A(1,0,0)@‘1,0,0),
AEC,
(24)
or, provided that t # 0, xF = tC, C + F = tA + A = (t-’ + x-‘)C and then ug =C(l,O,t) C,t E c, All partial (21),
(22),
@(~-‘+x-‘,o,l), I f 0. complex (24),
(25) conjugations
of vectors
A,sEC,
~;~=F(O,O,1)@(1,0,x),
FEC,
*2=D(0,1,0)@(0,1,0), 1’3
DEC,
ll;?
A
A(l,O,O)
U;2=C(l,o,t) c, t E c,
c3
(18),
(28) where projectors Po(@, correspond to the vectors a($) = (1/&)(1,ei@,e-2i~). Note that the integral representation (28) is not unique. The representation of the “dual” state ~2 is obtained by complex conjugation of projectors acting on the second space. For the case a = 0 we get simply the product state PO, (cf. ( 12) ) . Taking the parameter a arbitrarily close to 0, we obtain almost product pure states PQ, being nevertheless separable. The situation is, in a sense, analogical to the case of the states introduced in Ref. [ 121. The inseparability of the latter was also determined by a parametric change of both coherences and probabilities involved in the state. 4.2. 2 x 4 system
(25) are
~;~=A(l,s,O)@(l,s*,O),
=
which belongs just to Ran ea. Hence, for any c1 # 0, 1, the state c7;fi violates the condition due to the second statement of the theorem. Thus the state is inseparable together with the “dual” original state ea. In the latter the state PO, masks the inseparability due to einxp, making it “invisible” to the partial transposition criterion, but does not destroy it. It is interesting to see the limit behaviour of the state ea. In the case of a = 0 we get the separable state with the symmetric representation
(22)
For the case t # 0 we get from ( 19)) (20) j = 0 + F = 0 =+ E = 0 + D = 0. Hence the only product vector with nonvanishing z is the trivial zero vector. (iii) r # 0, s = 0. As in the case of (i) we can put r = 1 and A = A, B = B, c = C. Then we have B = E = D = 0 which leads to the equality
= (A,O.C;O,O,O;C
337
(1,&O),
@((P-l t # 0.
E
s # 0,
Here we will use the vectors i = 1,2,3,
(29)
C,
+x-‘,O,l),
O
(30)
(26) Then we can construct the following
state,
It is easy to see that the above vectors cannot span the range of @a as they are orthogonal to the vector (31) ti = (O,O, 1) @ (O,l,O).
(27)
i=l
which is inseparable (it can be easily verified like in the state ginxp using a partial transposition criterion). Now the states of our interest are of the form 7b cb
=
The corresponding
(32)
-by
7bfl
matrices are 3
bOO0
0
b0
0
ObOO OObO
0
Ob
0
0
00
b
OOOb
0
00
0
’ I 0000 bOO0 ObOO
;(l+b) 0 0
00&/s 60 Ob
0 0
OObO;J1_bZOO
u;*=F(O,l)@(l,O,O,y),
FEC,
u~*=D(1,0)~(0,0,0,1),
DEC.
bOO0 ObOO OObO OOOb
0 b 0 0
Let us check now whether the vectors (37) can be written in the above form. For the u;*, assuming that it is a nontrivial one (C f 0) and at the same time is of the form (38)) taking into account the coefficient Cl we obtain
;(l+b)
ObOO OObO OOOb
$(l+b) 0 0
00 00 60 Ob
0 0 0 0
OO;J1-b2 b0 0 Ob 0 ;(l+b)
I
(33)
It is easy to see that the state u:’ is positive as (34)
l\
J .
(37)
(38)
s*=s -1 .
0000~~~00
“=
(36)
A’, B’,C’, D’, E’ E C.
76-k 1
0010 0100 i 1000
s # 0,
w=(A’,B’,C’,D’;B’+yE’,C’,D’,E’),
1
/ooo
c, s E c,
On the other hand, any vectors from the range of a;f2 can be written in our notation as
(Tb=7b+1
X
of all possible product vectors ui E
1 +
~6nsep
clbT2 _--
plex conjugations Ran fib,
(35)
Consequently, for y = J(1 - b)/( 1 + b) f 0,l we get, analogically as Section 4.1, the partial com-
(39)
On the other hand, considering we have s* + ys(s*)-‘(1
+ y(s*>s)
B’, E’ and B’ + YE’,
= s(s*)?
(40)
Finally taking into account D’ we obtain (SW)-‘(l+y(s*)3)
=s.
(41)
Combining all the three equations above we find that ys* = 0, s Z 0 which contradicts the fact that y does not vanish. For uG2we obtain yE’ = D and at the same time E’ = yD, which is impossible for y # 0,l unless E’ = D = 0 trivialising then the vector u;~. For the vector u;* we get immediately that D = 0 must hold. It leads to the conclusion that none of vectors ur2 belongs to the Ran ~2 apart from the trivial zero one. Thus for any b # 0, 1 the state ffb violates our criterion from the theorem (statement (ii) ) being then inseparable together with its “dual” counterpart ~2. Here again the limit cases correspond to separable states. Namely we have go=s “PP , + $(Pc,c3e4+
pe,c4e,
1
.1The example of the pair of matrices of such a type treated, however, as operators on C4 63 C4 together with a similar analysis of their ranges has been considered in Ref. 191 in the context of positive maps.
(42)
P. Horodecki/
P/yic-.\ I.er~er.\A 232 ( 1997) .<33-339
where $(4) = (I/fi)(l,e’4) and F(4) = ( l/2) ( 1, eei6, ew2@, e-3i6). Putting b = 1 we obtain again the separable state Pp, . Thus we have provided the families of inseparable states with positive partial transposition. It is natural to ask how they are related to the necessary and sufficient separability condition given in terms of positive maps (Ref. [ 31, see Introduction). Clearly it follows that, in the presence of inseparability of states &, Ub, there must exist positive maps A, : L?(C3) ---f a(C3) and L&, : B(C4) + B(C2) such that the operators I @ Aa& and I @ &ffb are not positive, i.e., each of them possesses at least one negative eigenvalue. It is easy to see that the maps A,, Ah cannot be of the form A=Ay+AyT,
339
transposition criterion and they would also satisfy the present one as the latter is useful only for the states with a range essentially less than 7-L Now, as the set of separable states is closed, taking sufficiently small E one can ensure that the new states remain inseparable, despite their satisfying both criteria. The present criterion allowed us to provide examples of states of a new kind, where the entanglement is masked in a specific way by a classical admixture. In this context the interesting problem arises whether it is possible to distill such an entanglement using local operations and classical communication.
Acknowledgements
(43)
where &’ are completely positive maps and T is a transposition [ 31. However, the nature of the positive maps which are not of the form (43) is not known yet and finding the maps A,, Ab revealing the inseparability of the states pa, gb may be difficult.
5. Conclusion We have pointed out that any separable state can be written as a convex combination of only N pure product states ( N < (dim 31) 2, _We have provided a new necessary condition for separability of quantum states in terms of the range of density matrices. For any separable state it must be possible to span its range by a system of such product vectors that their counterparts obtained by partial complex conjugation span the range of partial transposition of the state. It is interesting to see that the above criterion sometimes does not reveal inseparability in cases where the partial transposition one works (this can be seen for the case of 2 x 2 states [ Ill) but it happens to be efficient where the latter fails. Thus, both the criteria are, in general, independent for mixed states, although one can easily verify (via Schmidt decomposition) their equivalence for pure states. One could suppose that, taken jointly, they can constitute the new necessary and sufficient condition of separability in higher dimensions. However, it is not the case [ 131: one can take the states a # 0,1, ((I-~)0-b+d/4,b # (1 -E)e,+E1/9, 0, 1). Those states will obviously satisfy the partial
The author is indebted to A. Sanpera for drawing his attention to the problem and for useful discussions. He also thanks T. Figiel and J. Popko for consultations on convex sets theory and R. Alicki, R. Horodecki and M. Horodecki for helpful comments. Special thanks are due to A. Peres for remarks leading to a significant improvement of the letter. This work is supported in part by the Polish Committee for Scientific Research, Contract No. 2 P03B 024 12. References [ I] C.H.
Bennett, D.P. Di Vincenzo, J. Smolin and W.K. Wootters, Phys. Rev. A 54 (1997) 3814. ]2] A. Penes, Phys. Rev. Lett. 76 (1997) 1413. [3] M. Horodecki, P Horodecki and R. Horodecki, Phys. Len. A 223 (1996) 1. [4] J.E Poyatos, J.I. Cirac and P Zoller, Phys. Rev. Lett. 78 (1997) 390. [ 51 V. Buick, V. Vedral, M.B. Plenio, PL. Knight and M. Hillery, Phys Rev. A 55 (1997) 3327. [6] M. Horodecki, P Horodecki and R. Horodecki, Phys. Rev. L&t. 78 (1997) 574. [ 71 PJ. Kelly and M.L. Weiss, Geometry and Convexity (John Wiley, New York, 1979). [ 81 E.M. Alfsen, Compact Convex Sets and Boundary Integrals ( Springer-Verlag, Berlin, 197 1) [9] S.L. Woronowicz, Rep. Math. Phys. 10 ( 1976) 165. [ IO] L.P Hughston, R. Jozsa and W.W. Wooters, Phys. Len. A 183 (1993) 14. [ 1I] R.E Werner, Phys. Rev. A 40 (1989) 4277. [ 121 R. Horodecki, Phys. Lett. A 210 (1996) 223. [ 131 A. Peres, private communication.