Entangled states close to the maximally mixed state
Résumé
This paper deals with the radius of the largest ball of separable mixed states around the maximally mixed state for multiqubit systems. This radius determines how close entangled states can be to the maximally mixed state. In Aubrun and Szarek (e-print arXiv:quant-ph/0503221) an upper bound on the radius was given, while in Gurvits and Barnum (e-print arXiv:quant-ph/0409095) a lower bound was provided. In this paper we improve both the upper and the lower bound, bringing the ratio of these bounds down to a constant c=√34∕27≈1.122, as opposed to a term of order √m log m for the best bounds known previously, where m is the number of qubits in the system. We construct concrete examples of separable states on the boundary to entanglement which realize the upper bounds. As a by-product, we compute the radii of the largest balls that fit into the projective tensor product of three and four unit balls in R3 and in the projective tensor product of an arbitrary number of unit balls in Rn for n=2, 4, and 8.