Robert Lockhart | Searcharxiv

arXiv2001

Low rank separable states are a set of measure zero within the set of low rank states

It is shown that the set of rank r separable states is measure zero within the set of low rank states provided r is less than an upper bound which depends upon the number of particles and the dimensions of the spaces they are modelled on. The upper bound is given. In the bipartite case in which both particles are modelled on m-dimensional hilbert space it is (m-1)^2. In the case of p qubits it is (2^p)-p. This paper is a corretion of one I recently posted and subsequently withdrew. That paper claimed the set of rank r separable states is measure zero if r is non-maximal rank. That may be true, but the proof in the withdrawn paper was false.

quant-ph↗

arXiv2000

Preserving entanglement under decoherence and sandwiching all separable states

Every entangled state can be perturbed, for instance by decoherence, and stay entangled. For a large class of pure entangled states, we show how large the perturbation can be. Our class includes all pure bipartite and all maximally entangled states. For an entangled state, E, the constucted neighborhood of entangled states is the region outside two parallel hyperplanes, which sandwich the set of all separable states. The states for which these neighborhoods are largest are the maximally entangled ones. As the number of particles, or the dimensions of the Hilbert spaces for two of the particles increases, the distance between two of the hyperplanes which sandwich the separable states goes to zero. It is easy to decide if a state Q is in the neighborhood of entangled states we construct for an entangled state E. One merely has to check if the trace of EQ is greater than a constant which depends upon E and which we determine.

quant-ph↗