Exact catalytic protocols for boundary unsteerable Gaussian states

Determine whether an exactly returned catalyst can generate remote Wigner negativity from a Gaussian state lying on the boundary of the unsteerable set, such as the lossy two-mode squeezed state at transmission one half.

Background

The no-catalysis theorem assumes strict unsteerability, which provides a full-support Gaussian smearing decomposition used in the proof. Boundary states do not satisfy this strict condition, and the smearing argument fails because Bob’s conditional state can be locked to Alice’s phase-space point. The authors explicitly state that neither a proof nor a counterexample is known for this boundary case.

References

We have neither a proof nor a counterexample for this case.

— Exact Catalysis Cannot Overcome the Gaussian-Steering Barrier for Remote Wigner Negativity  (2610.08509 - Chakrabarty, 6 Oct 2026) in Appendix, Section “Proof of Theorem~\ref{thm:nocat},” paragraph beginning “On the boundary”