Exact characterization of the gap between thermal operations and enhanced thermal operations

Determine the exact nature of the gap between thermal operations—channels implementable using a Gibbs-state ancilla and an energy-preserving unitary—and enhanced thermal operations—time-translation covariant operations that preserve the Gibbs state—beyond asymmetry manipulation. Specifically, characterize necessary and sufficient conditions that identify which enhanced thermal operations are not realizable as thermal operations and delineate the structural features responsible for this gap in general (non-Gaussian) settings.

Background

Thermal operations (TO) are defined operationally as channels implementable with an energy-preserving unitary and a Gibbs-state ancilla, while enhanced thermal operations (EnTO) are defined axiomatically as covariant operations that preserve the Gibbs state. In finite-dimensional settings, EnTO strictly contains TO, and this strict inclusion also holds for state transformations, indicating the presence of a genuine gap.

The paper proves that in the Gaussian regime the gap closes—Gaussian enhanced thermal operations coincide with Gaussian thermal operations—thus resolving the question for this important class. However, outside the Gaussian setting, the exact nature and structure of the gap remain unresolved, motivating a precise characterization of the difference between EnTO and TO beyond the known asymmetry-related origin.