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In this note we give sharp estimates on the volume of the set of separablestates on N qubits. In particular, the magnitude of the "effective radius" ofthat set in the sense of volume is determined up to a factor which is a (small)power of N, and thus precisely on the scale of powers of its dimension.Additionally, one of the appendices contains sharp estimates (by known methods)for the expected values of norms of the GUE random matrices. We employ standardtools of classical convexity, high-dimensional probability and geometry ofBanach spaces.Comment: 20 p., LATEX; an expanded version of the original submission: more  background material from convexity and geometry of Banach spaces, more  exhaustive bibliography and improved quality of references to the  bibliograph

Volume of separable states is super-doubly-exponentially small in the number of qubits