qu an t - ph / 9 60 50 38 v 2 2 6 Ju n 96 Separability of mixed states : necessary and sufficient conditions

We provide necessary and sufficient conditions for separability of mixed states. As a result we obtain a simple criterion of separability for 2 × 2 and 2× 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. However, it is not the case in general. Some examples of mixtures which demonstrate the utility of the criterion are considered. Typeset using REVTEX ∗e-mail: [email protected] 1 Quantum inseparability, first recognized by Einstein, Podolsky and Rosen [1] and Schrödinger [2], is one of the most astonishing features of quantum formalism. After over sixty years it is still a fascinating object from both theoretical and experimental points of view. Recently, together with a dynamical development of experimental methods, a number of possible practical applications of the quantum inseparable states has been proposed including quantum computation [3] and quantum teleportation [4]. The above ideas are based on the fact that the quantum inseparability implies, in particular, the existence of the pure entangled states which produce nonclassical phenomena. However, in laboratory one deals with mixed states rather than pure ones. This is due to the uncontrolled interaction with the enviroment. Then it is very important to know which mixed states can produce quantum effects. The problem is much more complicated than in the pure states case. It may be due to the fact that mixed states apparently possess the ability to behave classically in some respects but quantum mechanically in others [5]. In accordance with the so-called generalized inseparability principle [6] we will call a mixed state of a compound quantum system inseparable if it cannot be written as a convex combination of product states. The problem of the inseparability of mixed states was first raised by Werner [7], who constructed a family of inseparable states which admit the local hidden variable model. It has been pointed out [8] that, nevertheless, some of them are nonlocal and this “hidden” nonlocality can be revealed by subjecting them to more complicated experiments than single von Neumann measurements considered by Werner (see also Ref. [9] in this context). This shows that it is hard to classify the mixed states as purely classical or quantum ones. Recently the separable states have been investigated within the information-theoretic approach [6, 10, 11, 12]. It has been shown that they satisfy a series of the so-called quantum α-entropy inequalities (for α = 1, 2 [10, 11] and α = ∞ [12]). Moreover, the separable two spin2 states with maximal entropies of subsystems have been completely characterized in terms of the α-entropy inequalities [6]. It is remarkable that there exist inseparable states which do not reveal nonclassical features under the entropic criterion [11]. 2 Then the fundamental problem of an “operational” characterization of the separable states arises. So far only some necessary conditions of separability have been found [6, 7, 10, 11, 13]. An important step is due to Peres [14], who has provided a very strong condition. Namely, he noticed that the separable states remain positive if subjected to partial transposition. Then he conjectured that this is also sufficient condition. In this Letter we present two necessary and sufficient conditions for separability of mixed states. It provides a complete, operational characterization of separable states for 2× 2 and 2 × 3 systems. It appears that Peres’ conjecture is valid for those cases. However, as we show in the Appendix, the conjecture is not valid in general. We also illustrate our results by means of some examples and discuss them in the context of the α-entropy inequalities. To make our considerations more clear, we start from the following notation and definitions. We will deal with the states on the finite dimensional Hilbert space H = H1 ⊗H2. An operator % acting on H is a state if Tr% = 1 and if it is a positive operator i.e. Tr%P ≥ 0 (1) for any projectors P . A state is called separable 1 if it can be approximated in the trace norm by the states of the form % = k ∑ i=1 pi%i ⊗ %̃i (2) where %i and %̃i are states on H1 and H2 respectively. By A1 and A2 we will denote the set of the operators acting on H1 and H2 respectively. Recall that Ai constitute a Hilbert space (so-called Hilbert-Schmidt space) with scalar product 〈A,B〉 = TrB†A. The space of the linear maps from A1 to A2 is denoted by L(A1,A2). We say that a map Λ ∈ L(A1,A2) is positive if it maps positive operators in A1 into the set of positive operators i.e. if A ≥ 0 implies Λ(A) ≥ 0. Finally we need the definition of completely positive map. One says [15] that a map Λ ∈ L(A1,A2) is completely positive if the induced map The presented definition of separable states is due to Werner [7] who called them classically correlated states. 3 Λn = Λ⊗ I : A1 ⊗Mn → A2 ⊗Mn (3) is positive for all n. Here Mn stand for the set of the complex matrices n× n and I is the identity map. Thus the tensor product of a completely positive map and the identity maps positive operators into positive ones. It is remarkable that there are positive maps that do not possess this property. This fact is crucial for the problem we discuss here. Indeed, trivially, the product states are mapped into positive operators by the tensor product of a positive map and identity: (Λ⊗ I)%⊗ %̃ = (Λ%)⊗ %̃ ≥ 0. Of course, the same holds for the separable states. Then our main idea is that this property of the separable states is essential i.e., roughly speaking, if a state % is inseparable, then there exists a positive map Λ such that Λ⊗ I% is not positive. This means that we can recognize the inseparable states by means of the positive maps. Now the point is that not all the positive maps can help us to determine whether a given state is inseparable. In fact, the completely positive maps do not “feel” the inseparability. Thus the problem of characterization of the set of the separable states reduces to the following: one should extract from thet set of all positive maps some essential ones. As we will see further, it is possible in some cases. Namely it appears that for the 2× 2 and 2× 3 systems the transposition is the only such map. We will start from the following Lemma 1 For any inseparable state %̃ ∈ A1 ⊗ A2 there exists Hermitian operator à such that Tr(Ã%̃) < 0 and Tr(Ãσ) ≥ 0 (4) for any separable σ. Proof.-From the definition of the set of separable states it follows that it is both convex and closed set in A1⊗A2. Thus we can apply a theorem (conclusion from the Hahn-Banach 2Of course a completely positive map is also a positive one. 4 theorem) [16] which, for our purposes, can be formulated as follows. If W1, W2 are convex closed sets in a real Banach space and one of them is compact, then there exists a continous functional f and α ∈ R such that for all pairs w1 ∈W1, w2 ∈W2 we have f(w1) < α ≤ f(w2) (5) This theorem says, in particular, that a closed convex set in the Banach space is completely described by the inequalities involving continous functionals. Noting that one-element set is compact we obtain that there exists a real functional g on the real space à generated by Hermitian operators from A1 ⊗A2 such that g(%̃) < β ≤ g(σ), (6) for all separable σ. It is a well known fact that any contionous functional g on a Hilbert space can be represented by a vector from this space. As à is a (real) Hilbert space we obtain that the functional g can be represented as g(%) = Tr(%A), (7) where A = A†. Now, if I stands for identity then for any states %, σ one has obviously Tr(βI%) = Tr(βIσ) = β. Thus it sufficies only to take