This paper deals with estimating the volume of the set of separable mixed quantum states when the dimension of the state space grows to infinity. This has been studied recently for qubits; here we consider larger particles and conclude that, in all cases, the proportion of the states that are separable is superexponentially small in the dimension of the set. We also show that the partial transpose criterion becomes imprecise when the dimension increases, and that the lower bound 6 −N/2 on the ͑Hilbert-Schmidt͒ inradius of the set of separable states on N qubits obtained recently by Gurvits and Barnum is essentially optimal. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces. One relatively nonstandard point is a formal introduction of the concept of projective tensor products of convex bodies, and an initial study of this concept.

Abstract. Let W be a Wishart random matrix of size d 2 × d 2 , considered as a block matrix with d × d blocks. Let Y be the matrix obtained by transposing each block of W . We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d → ∞. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method.This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C d ⊗ C d obtained after partial tracing a random pure state over some ancilla of dimension αd 2 is typically PPT when α > 4 and typically non-PPT when α < 4.