Bipartite-mixed-states of infinite-dimensional systems are generically nonseparable

Given a bipartite quantum system represented by a Hilbert space ${\mathcal{H}}_1\otimes {\mathcal{H}}_2,$ we give an elementary argument to show that if either $\dim {\mathcal{H}}_1=\infty$ or $\dim {\mathcal{H}}_2=\infty ,$ then the set of nonseparable density operators on ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$ is trace-norm dense in the set of all density operators (and the separable density operators nowhere dense). This result complements recent detailed investigations of separability, which show that when $\dim {\mathcal{H}}_i<\infty$ for $i=1,2,$ there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.