The volume of separable states is super-doubly-exponentially small

Abstract

In this note we give sharp estimates on 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. Additionally, one of the appendices contains sharp estimates (by known methods) for the expected values of norms of the GUE random matrices. We employ standard tools of classical convexity, high-dimensional probability and geometry of Banach spaces.

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 bibliography

References in corpus (13)

• Level-Spacing Distributions and the Airy Kernel
• On the volume of the set of mixed entangled states
• On Orthogonal and Symplectic Matrix Ensembles
• On the role of entanglement in quantum computational speed-up
• Robustness of entanglement
• Separability of very noisy mixed states and implications for NMR quantum computing
• On the volume of the set of mixed entangled states II
• Bures volume of the set of mixed quantum states
• Hilbert--Schmidt volume of the set of mixed quantum states
• Separable balls around the maximally mixed multipartite quantum states
• Qudit Entanglement
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• Complete Separability and Fourier representations of n-qubit states

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