Physical implementation of compatibility-domain selection for non-completely-positive dynamics

Construct an explicit and formal physical mechanism that selects the compatibility domain of initial system–ancilla states on which a non-completely-positive reduced map and its ancilla extensions remain positive, including mechanisms such as a dynamical preparation or transient slippage of initial conditions.

Background

The Pechukas–Alicki debate considers whether non-completely-positive reduced maps can be physically meaningful when the initial system–environment state is restricted. For a non-completely-positive map, one may restrict the action of the extended map to a compatibility domain consisting of initial states that remain positive after evolution.

The paper notes that identifying this domain is not enough: a physical process must explain how harmless initial states are selected. Although slippage-of-initial-conditions models have been proposed, the text leaves open the availability of explicit formal implementations of such selection mechanisms.

References

However, explicit and formal implementations of the selection of harmless initial conditions do not seem to be available, yet.

— Fifty Years of the GKLS Master Equation: Foundations and Early Developments  (2609.19404 - Benatti et al., 16 Sep 2026) in Section 4, subsection “Pechukas: challenging complete positivity (1994)”

Finally, since entanglement becomes typical with increasing dimension, one suspects that, for \Lambda_t NCP when d\gg1, the compatibility domains for \Lambda_t\otimes{\rm id}_k on S_d+S_d may become thinner and thinner.

— Fifty Years of the GKLS Master Equation: Foundations and Early Developments  (2609.19404 - Benatti et al., 16 Sep 2026) in Section 4, subsection “Pechukas: challenging complete positivity (1994)”