Separability from Spectrum for Qubit-Qudit States

TL;DR

This paper characterizes when bipartite qubit-qudit states are separable from spectrum, showing that for all such states, being PPT under all unitaries is equivalent to being separable from spectrum, extending previous results.

Contribution

It completely solves the separability from spectrum problem for all qubit-qudit states, establishing a new equivalence with positive partial transpose across all unitaries.

Findings

Separable from spectrum states are PPT under all unitaries.

The equivalence between separability from spectrum and PPT is established for all qubit-qudit states.

Contrasts with the usual PPT criterion where PPT is weaker than separability.

Abstract

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 <= 3. We solve all remaining qubit-qudit cases (i.e., when m = 2 and n >= 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.