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

Journal Article · Thu Sep 15 00:00:00 EDT 2005 · Physical Review. A

DOI:https://doi.org/10.1103/PhysRevA.72.032304· OSTI ID:20718490

Szarek, Stanislaw J ^[1]

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1. Equipe d'Analyse Fonctionnelle, B.C. 186, Universite Paris VI, 4, Place Jussieu, F-75252 Paris (France) and Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106 (United States)

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.

• OSTI ID: 20718490
• Journal Information: Physical Review. A, Vol. 72, Issue 3; Other Information: DOI: 10.1103/PhysRevA.72.032304; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

74 ATOMIC AND MOLECULAR PHYSICS
ALGEBRA
BANACH SPACE
GEOMETRY
MATRICES
PROBABILITY
QUANTUM COMPUTERS
QUANTUM ENTANGLEMENT
QUANTUM MECHANICS
QUBITS
RANDOMNESS