On the necessary and sufficient conditions for separability of mixed quantum states

A state acting on Hilbert space H1 ⊗ H2 is called separable if it can be approximated in trace norm by convex combinations of product states. We provide necessary and sufficient conditions for separability of mixed states in terms of functionals and positive maps. As a result we obtain a complete characterization of separable states for 2 × 2 and 2 × 3 systems. Here, the positivity of the partial transposition of a state is necessary and sufficient for its separability. Typeset using REVTEX ∗e-mail: fizrh@univ.gda.pl 1 Quantum inseparability, first recognized in 1935 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 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 environment. 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. In accordance with the so-called generalized inseparability principle [5] we will call a mixed state of compound quantum system inseparable if it cannot be written as convex combination of product states. The problem of inseparability of mixed states was first raised by Werner [6], who constructed a family of inseparable states which admit the local hidden variable model. It has been pointed pointed out [7] 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. This shows that it is hard to divide the mixed states into definitely quantum and classical ones. Recently the separable states have been investigated within the information-theoretic approach [5,8–10]. It has been shown that they satisfy a series of the so-called quantum α-entropy inequalities (for α = 1, 2 [8,9] and α = ∞ [10]). Moreover, the separable two spin2 states with maximal entropies of subsystems have been completely characterized in terms of α-entropy inequalities [5]. It is remarkable that there exist inseparable states which do not reveal nonclassical features under the entropy criterion [9]. 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 [5,6,8,9,11]. An important step is due to Peres [12], 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 paper 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. 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.