Separability from spectrum for qubit-qudit states
Nathaniel Johnston · Physical Review A · 2013
The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states $\ensuremath{\rho}$ with the property that ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ is separable for all unitary matrices $U$. This problem has been solved when the local dimensions $m$ and $n$ satisfy $m=2$ and $n\ensuremath{\le}3$. We solve all remaining qubit-qudit cases (i.e., when $m=2$ and $n\ensuremath{\ge}4$ is arbitrary). In all of these cases we show that a state is separable from spectrum if and only if ${U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\rho}U$ has positive partial transpose for all unitary matrices $U$. This equivalence is in stark contrast with the usual separability problem, where a state having positive partial transpose is a strictly weaker property than it being separable.