Transfer rapid-mixing results to the finite erf filter

Derive the transfer argument needed to apply rapid-mixing results for related one-sided filtered-jump constructions to the finite erf-discretised filtered-jump generator used for dissipative ground-state preparation.

Background

The stationary-state guarantee for truncated filtered jumps is conditional on a mixing pair for the corresponding global Lindbladian. The paper notes that rapid mixing has been proved for a related one-sided filtered-jump construction in weakly interacting systems, but that the result does not directly apply to the finite erf-discretised filter studied here.

Applying the existing rapid-mixing theorem would therefore require a rigorous transfer argument connecting the related construction to the finite discretised generator. The paper explicitly states that this argument has not been carried out, leaving the applicability of those mixing results unresolved.

References

For weakly interacting systems, where cluster expansions already make the Gibbs state classically tractable at low, high and, more recently, arbitrary temperature, rapid mixing has been proved for a related one-sided filtered-jump construction; using that result here would need a transfer argument to our finite erf discretisation, which we have not carried out.

— Locality and filter design for dissipative ground-state preparation  (2609.31266 - Elman, 25 Sep 2026) in Section Discussion (Section 4), paragraph beginning “The third observation, that convergence survives localisation”