Structure of the thermodynamic arrow of time in classical and quantum theories

TL;DR

This paper investigates the structure of the thermodynamic arrow of time in classical and quantum systems, revealing a lattice structure at infinite temperature and its preservation in simple quantum cases at finite temperatures.

Contribution

It demonstrates the lattice structure of the thermodynamic arrow at infinite temperature and shows quantum coherence can preserve this structure at finite temperatures.

Findings

Lattice structure exists at infinite temperature in both classical and quantum regimes.

Classical lattice structure breaks at finite temperature, but quantum case preserves it.

Quantum coherence may be essential for optimal history erasure processes.

Abstract

In this work we analyse the structure of the thermodynamic arrow of time, defined by transformations that leave the thermal equilibrium state unchanged, in classical (incoherent) and quantum (coherent) regimes. We note that in the infinite-temperature limit the thermodynamic ordering of states in both regimes exhibits a lattice structure. This means that when energy does not matter and the only thermodynamic resource is given by information, the thermodynamic arrow of time has a very specific structure. Namely, for any two states at present there exists a unique state in the past consistent with them and with all possible joint pasts. Similarly, there also exists a unique state in the future consistent with those states and with all possible joint futures. We also show that the lattice structure in the classical regime is broken at finite temperatures, i.e., when energy is a relevant thermodynamic resource. Surprisingly, however, we prove that in the simplest quantum scenario of a two-dimensional system, this structure is preserved at finite temperatures. We provide the physical interpretation of these results by introducing and analysing the history erasure process, and point out that quantum coherence may be a necessary resource for the existence of an optimal erasure process.