Equality of absolutely separable and absolutely PPT sets in qutrit–qudit systems

Determine whether the set of absolutely separable states equals the set of absolutely positive partial transpose (PPT) states for bipartite qutrit–qudit systems M_3 ⊗ M_n for all n ≥ 2.

Background

In qubit–qudit systems (M_2 ⊗ M_n), it is known that the sets of absolutely separable and absolutely PPT states coincide, and there exists a simple spectral criterion characterizing absolute separability. In contrast, for higher local dimensions, especially the qutrit–qudit setting, explicit characterizations are more involved and only absolute PPT conditions are known (e.g., Hildebrand’s criterion).

Clarifying whether absolute separability and absolute PPT coincide in the qutrit–qudit case would settle a fundamental question in the separability-from-spectrum program and guide the search for spectral conditions that characterize absolute separability beyond the qubit–qudit regime.

References

It remains unknown whether the set of absolutely separable states coincides with the set of absolutely PPT states for the qutrit-qudit system .

— On the maximum purity of absolutely separable bipartite states  (2510.19508 - Dung et al., 22 Oct 2025) in Section 2 (Preliminaries)

In higher bipartite dimensions, whether the two sets coincide remains open .

— Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay  (2609.18568 - Tran, 16 Sep 2026) in Section 1, Introduction and main results

Despite this progress, whether the two notions coincide remains an open problem. In other words, it is not known whether

\mathrm{APPT}{d_1,d_2}=\mathrm{AS}{d_1,d_2} \quad \text{for all } d_1,d_2\geq3,

where $APPT_{d_1,d_2}$ and $\mathrm{AS}_{d_1,d_2}$ stand for the sets of absolutely PPT and absolutely separable states in $d_1\otimes d_2$, respectively.

— On the separability of Bell-diagonal states with absolutely positive partial transposition  (2610.12000 - Patra et al., 8 Oct 2026) in Section 1, Introduction