Bipartite-mixed-states of infinite-dimensional systems are generically nonseparable
Rob Clifton, Hans Halvorson · Physical Review A · 1999
Given a bipartite quantum system represented by a Hilbert space ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2},$ we give an elementary argument to show that if either $\mathrm{dim}{\mathcal{H}}_{1}=\ensuremath{\infty}$ or $\mathrm{dim}{\mathcal{H}}_{2}=\ensuremath{\infty},$ then the set of nonseparable density operators on ${\mathcal{H}}_{1}\ensuremath{\bigotimes}{\mathcal{H}}_{2}$ 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 $\mathrm{dim}{\mathcal{H}}_{i}<\ensuremath{\infty}$ for $i=1,2,$ there is a separable neighborhood (perhaps very small for large dimensions) of the maximally mixed state.