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This note deals with estimating the volume of the set of separable mixedquantum states when the dimension of the state space grows to infinity. Thishas been studied recently for qubits; here we consider larger particles andconclude that, in all cases, the proportion of the states that are separable issuper-exponentially small in the dimension of the set. We also show that thepartial transpose criterion becomes imprecise when the dimension increases, andthat the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set ofseparable states on N qubits obtained recently by Gurvits and Barnum isessentially optimal. We employ standard tools of classical convexity,high-dimensional probability and geometry of Banach spaces. One relativelynon-standard point is a formal introduction of the concept of projective tensorproducts of convex bodies, and an initial study of this concept.  PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft, 02.50.Cw  MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68Comment: 15 pages; this version includes minor changes in exposition; no new  results but some numerical constants improve

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

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