Thermal thresholds for larger classes of states

Determine thermal entanglement-death thresholds when fully separable states are replaced by larger state classes, including convex mixtures of states prepared by shallow circuits or circuits lying in a fixed level of the magic hierarchy.

Background

The main theorem characterizes the threshold for decomposition into mixtures of product Pauli eigenstates, a subset of fully separable states. The paper explicitly raises the unresolved extension to broader classes that can contain structured entanglement or nonstabilizer resources, such as shallow-circuit states and fixed magic-hierarchy circuit classes.

References

Several open questions remain. The sharpness construction allows the locality of a term to grow with $\Delta$, so the optimal constants at a fixed locality are undetermined (e.g. say $k=2$ local Hamiltonians). One can also ask for thermal thresholds relative to larger classes than product states, such as convex mixtures of states prepared by shallow circuits or circuits in a fixed level of the magic hierarchy.

— Sharp universal death of entanglement threshold for Pauli Hamiltonians  (2609.30149 - Kiani, 24 Sep 2026) in Section Discussion