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

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 new 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.

[1]  P. Slater.  The silver mean and volumes of the separable two-qubit states , 2003 .

[2]  H. Sommers,et al.  Bures volume of the set of mixed quantum states , 2003, quant-ph/0304041.

[3]  L. Gurvits,et al.  Separable balls around the maximally mixed multipartite quantum states , 2003, quant-ph/0302102.

[4]  R. Jozsa,et al.  On the role of entanglement in quantum-computational speed-up , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[5]  Uniwersytet Jagiello,et al.  Hilbert-Schmidt volume of the set of mixed quantum states , 2003 .

[6]  S. Thorbjørnsen,et al.  Random matrices with complex Gaussian entries , 2003 .

[7]  Michel Ledoux,et al.  A Remark on Hypercontractivity and Tail Inequalities for the Largest Eigenvalues of Random Matrices , 2003 .

[8]  M. Horodecki,et al.  Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments , 2002 .

[9]  M. Horodecki,et al.  Mixed-State Entanglement and Quantum Communication , 2001, quant-ph/0109124.

[10]  A. O. Pittenger,et al.  Convexity and the separability problem of quantum mechanical density matrices , 2001, quant-ph/0103038.

[11]  G. Milburn,et al.  Qudit Entanglement , 2000, quant-ph/0001075.

[12]  Complete separability and Fourier representations of n-qubit states , 1999, quant-ph/9912116.

[13]  R. Jozsa,et al.  SEPARABILITY OF VERY NOISY MIXED STATES AND IMPLICATIONS FOR NMR QUANTUM COMPUTING , 1998, quant-ph/9811018.

[14]  M. Lewenstein,et al.  On the volume of the set of mixed entangled states II , 1999, quant-ph/9902050.

[15]  C. Tracy,et al.  On orthogonal and symplectic matrix ensembles , 1995, solv-int/9509007.

[16]  M. Ledoux,et al.  Isoperimetry and Gaussian analysis , 1996 .

[17]  S. Braunstein,et al.  Statistical distance and the geometry of quantum states. , 1994, Physical review letters.

[18]  C. Tracy,et al.  Level-spacing distributions and the Airy kernel , 1992, hep-th/9210074.

[19]  B. Bollobás.  THE VOLUME OF CONVEX BODIES AND BANACH SPACE GEOMETRY (Cambridge Tracts in Mathematics 94) , 1991 .

[20]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[21]  V. Milman,et al.  New volume ratio properties for convex symmetric bodies in ℝn , 1987 .

[22]  Jean Saint Raymond,et al.  Le volume des idéaux d'opérateurs classiques , 1984 .

[23]  Nicole Tomczak-Jaegermann,et al.  On nearly euclidean decomposition for some classes of Banach spaces , 1980 .

[24]  G. C. Shephard,et al.  Convex Bodies Associated with a Given Convex Body , 1958 .

[25]  G. C. Shephard,et al.  The difference body of a convex body , 1957 .