Necessary and sufficient criterion for absolute separability beyond the qubit–qudit case

Develop a necessary and sufficient spectral criterion for the absolute separability of bipartite mixed states in dimensions beyond M_2 ⊗ M_n (i.e., for general m ⊗ n with m ≥ 3), thereby extending the separability-from-spectrum characterization to higher local dimensions.

Background

In the qubit–qudit setting, absolute separability coincides with absolute PPT and admits a simple spectral inequality characterizing when a state is absolutely separable. For higher dimensions, only partial characterizations are known—most notably for absolute PPT—while an exact necessary and sufficient criterion for absolute separability is missing.

Establishing such a criterion would be central to understanding the geometry of absolutely separable states, their relationship to absolute PPT, and the broader separability-from-spectrum program across all bipartite dimensions.