Entangled states close to the maximally mixed state

Abstract

We give improved upper bounds on the radius of the largest ball of separable states of an m-qubit system around the maximally mixed state. The ratio between the upper bound and the best known lower bound (Hildebrand, quant.ph/0601201) thus shrinks to a constant c = \sqrt{34/27} ~ 1.122, as opposed to a term of order \sqrt{m\log m} for the best upper bound known previously (Aubrun and Szarek, quant.ph/0503221). We give concrete examples of separable states on the boundary to entanglement which realize these upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of four unit balls in R^3 and in the projective tensor product of an arbitrary number of unit balls in R^n for n = 2,4,8.

11 pages; v2: n qubit case added

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• A bipartite separable ball and its applications
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• Concurrence of Lorentz-positive maps
• Separable ellipsoids around multipartite states
• Applications of differential geometry and representation theory to description of quantum correlations
• On the extreme points of sets of absolulely separable and PPT states
• Maximum Dimension of Subspaces with No Product Basis
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• Properties of Local Quantum Operations with Shared Entanglement