Separability from Spectrum for Qubit-Qudit States


The separability from spectrum problem asks for a characterization of the eigenvalues of the bipartite mixed states \rho with the property that U^\dagger\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^\dagger\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.


  • Nathaniel Johnston


Cite as:

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

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  1. Nirman Ganguly
    March 23rd, 2014 at 01:06 | #1

    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.

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