1,985 research outputs found
Better bound on the exponent of the radius of the multipartite separable ball
We show that for an m-qubit quantum system, there is a ball of radius
asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at
the identity matrix, of separable (unentangled) positive semidefinite matrices,
for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much
smaller in magnitude than the best previously known exponent, from our earlier
work, of 1/2. For normalized m-qubit states, we get a separable ball of radius
sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times
6^{-m/2} (note that \kappa = \sqrt{3}), compared to the previous 2 * 2^{-3m/2}.
This implies that with parameters realistic for current experiments, NMR with
standard pseudopure-state preparation techniques can access only unentangled
states if 36 qubits or fewer are used (compared to 23 qubits via our earlier
results). We also obtain an improved exponent for m-partite systems of fixed
local dimension d_0, although approaching our earlier exponent as d_0
approaches infinity.Comment: 30 pp doublespaced, latex/revtex, v2 added discussion of Szarek's
upper bound, and reference to work of Vidal, v3 fixed some errors (no effect
on results), v4 involves major changes leading to an improved constant, same
exponent, and adds references to and discussion of Szarek's work showing that
exponent is essentially optimal for qubit case, and Hildebrand's alternative
derivation for qubit case. To appear in PR
Isomorphic properties of Intersection bodies
We study isomorphic properties of two generalizations of intersection bodies,
the class of k-intersection bodies and the class of generalized k-intersection
bodies. We also show that the Banach-Mazur distance of the k-intersection body
of a convex body, when it exists and it is convex, with the Euclidean ball, is
bounded by a constant depending only on k, generalizing a well-known result of
Hensley and Borell. We conclude by giving some volumetric estimates for
k-intersection bodies
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