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

Abstract

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 That may be true, but the proof in the withdrawn paper was false.

3 pages

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