The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states $\rho$ with the property that $U^{†}\rho U$ is separable for all unitary matrices $U$. This problem has been solved when the local dimensions $m$ and $n$ satisfy $m=2$ and $n\le 3$. We solve all remaining qubit-qudit cases (i.e., when $m=2$ and $n\ge 4$ is arbitrary). In all of these cases we show that a state is separable from spectrum if and only if $U^{†}\rho U$ 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.

• Received 16 September 2013

DOI:https://doi.org/10.1103/PhysRevA.88.062330