Role of Dilations in Reversing Physical Processes: Tabletop Reversibility and Generalized Thermal Operations

TL;DR

This paper explores the fundamental role of dilations in reversing physical processes, establishing the equivalence of two definitions of reverse channels, and characterizing classes of reversible maps, especially in quantum systems.

Contribution

It proves the equivalence of Bayesian retrodiction and dilation-based reverse channels in classical and quantum settings, and characterizes reversible classes like product-preserving and thermal maps.

Findings

Bayesian and dilation-based reversals are equivalent when system-bath correlations are considered.

Product-preserving maps are not always reversible with the same devices, but are sufficient for tabletop reversibility.

Preservation of local energy spectra characterizes generalized thermal operations.

Abstract

Irreversibility, crucial in both thermodynamics and information theory, is naturally studied by comparing the evolution -- the (forward) channel -- with an associated reverse -- the reverse channel. There are two natural ways to define this reverse channel. Using logical inference, the reverse channel is the Bayesian retrodiction (the Petz recovery map in the quantum formalism) of the original one. Alternatively, we know from physics that every irreversible process can be modeled as an open system: one can then define the corresponding closed system by adding a bath ("dilation"), trivially reverse the global reversible process, and finally remove the bath again. We prove that the two recipes are strictly identical, both in the classical and in the quantum formalism, once one accounts for correlations formed between system and the bath. Having established this, we define and study special classes of maps: product-preserving maps (including generalized thermal maps), for which no such system-bath correlations are formed for some states; and tabletop time-reversible maps, when the reverse channel can be implemented with the same devices as the original one. We establish several general results connecting these classes, and a very detailed characterization when both the system and the bath are one qubit. In particular, we show that when reverse channels are well-defined, product-preservation is a sufficient but not necessary condition for tabletop reversibility; and that the preservation of local energy spectra is a necessary and sufficient condition to generalized thermal operations.