ar X iv : q ua nt - p h / 05 09 00 8 v 1 1 S ep 2 00 5 On the structure of the body of states with positive partial transpose

We show that the convex set of separable mixed states of the 2 × 2 system is a body of a constant height. This fact is used to prove that the probability to find a random state to be separable equals 2 times the probability to find a random boundary state to be separable, provided the random states are generated uniformly with respect to the Hilbert– Schmidt (Euclidean) measure. An analogous property holds for the set of positive-partial-transpose states for an arbitrary bipartite system. The phenomena of quantum entanglement became crucial in recent development of quantum information processing. In general it is not simple to decide whether a given mixed quantum state is entangled or separable [1]. The situation gets simpler for the qubit–qubit and the qubit–qutrit systems, for which it is known that a state is separable if and only if its partial transpose is positive [2, 3]. However, even in the simplest case of the two–qubit system, the structure of the set of separable states is not fully understood. This 15-dimensional convex set M S is known to contain the maximal ball inscribed in the set of all mixed states M(4) [4]. Although some work has been done to estimate the volume of the set of separable states [5, 6, 7, 8, 9] and to describe its geometry [10, 11], the exact volume of the set of separable states is unknown even in this simplest case [12]. In order to elucidate properties of the set of separable states Slater studied the probability that a random state be separable, inside the set of mixed states and at its boundary, using the Hilbert–Schmidt measure. For the two–qubit system he found numerically that the ratio Ω between these two probabilities is close to 2 [13, 14]. The fact that it is more likely to find a separable state