## Separability from Spectrum for Qubit-Qudit States

**Abstract:**

The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states with the property that 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 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.

**Authors:**

- Nathaniel Johnston

**Download:**

- Official publication from Physical Review A
- Preprint from arXiv:1309.2006 [quant-ph]
- Local preprint [pdf]
- Slideshow presentation (short version) [pdf]
- Slideshow presentation (long version) [PowerPoint .ppt]

**Cite as:**

- N. Johnston. Separability from spectrum for qubit–qudit states.
*Physical Review A*, 88:062330, 2013.

**Related publications:**

- Is absolute separability determined by the partial transpose? – a paper that considers the same problem in systems with arbitrary local dimensions

Dear Dr. Johnston,

I have read with interest your article on the separability from spectrum problem for qubit-qudit states. In this context I would appreciate your attention to our recent article[arxiv 1401.5324(quant-ph)] on a different characterization of the set of absolutely separable states using the Hahn-Banach theorem from functional analysis,which sheds some more light on the structure of states that are always separable from spectrum.