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

In this paper we give sharp two-sided estimates of the volume of the set of separable states on $N$ qubits. In particular, the magnitude of the “effective radius” of that 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. We also identify an ellipsoid that appears to optimally approximate the set of separable states. Additionally, one of the appendixes contains sharp estimates (by known methods) for the expected values of norms of the Gaussian unitary ensemble random matrices. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces.