Thermodynamics of quantum processes: An operational framework for free energy and reversible athermality

TL;DR

This paper extends thermodynamic free energy to quantum processes by defining a channel free energy F^β[𝒩] = β^{-1}D[𝒩 || 𝒯^β] relative to an absolutely thermal channel 𝒯^β, and builds a dynamical resource theory of athermality under Gibbs-preserving superchannels. It establishes operational meanings for one-shot and asymptotic channel distillation and formation, proving asymptotic reversibility with distillation and formation rates both equal to (1/2)D[𝒩 || 𝒯^β], and links these thermodynamic quantities to information-processing tasks such as private randomness, purity, erasure, and work extraction. The work shows that the resource-theoretic and thermal free energies of channels connect to extractable work via partial thermalization and relate to channel capacity through mutual information, providing a unified framework tying free energy, energy, entropy, and maximal extractable work to quantum-information processing capabilities. Overall, the framework offers a rigorous, axiomatic, and operational account of the thermodynamics of quantum processes, with implications for quantum engines, communication, and the foundational understanding of dynamical resources in quantum thermodynamics.

Abstract

We explore the thermodynamics of quantum processes (quantum channels) by axiomatically introducing the free energy for channels, defined via the quantum relative entropy with an absolutely thermal channel whose fixed output is in equilibrium with a thermal reservoir. This definition finds strong support through its operational interpretations in designated quantum information and thermodynamic tasks. We construct a resource theory of athermality for quantum processes, where free operations are Gibbs preserving superchannels and golden units are unitary channels with respect to absolutely thermal channel having fully degenerate output Hamiltonian. We exactly characterize the one-shot distillation and formation of quantum channels using hypothesis-testing and max-relative entropy with respect to the absolutely thermal channel. These rates converge asymptotically to the channel free energy (up to a multiplicative factor of half the inverse temperature), establishing its operational meaning and proving the asymptotic reversibility of the athermality. We show the direct relation between the resource theory of athermality and quantum information tasks such as private randomness and purity distillation, and thermodynamic tasks of erasure and work extraction. Our work connects the core thermodynamic concepts of free energy, energy, entropy, and maximal extractable work of quantum processes to their information processing capabilities.

Figures (1)

• Figure 1: A bipartite system with a non-interacting Hamiltonian $\widehat{H}_R+ \widehat{H}_A$ is in state $\psi_{RA'}$, in contact with a bath at inverse temperature $\beta$. A quantum channel $\mathcal{N}_{A'\to A}$ acts locally on $A'$ such that $\widehat{H}^{\rm int}_{RA}=0$. To extract work, we partially thermalize the channel output, $\mathcal{N}_{A'\to A}(\psi_{RA'})\to \psi_{R}\otimes \gamma_A$. We consider partial thermalization composed of three steps: (1) Decoupling: $\mathcal{N}(\psi_{RA'})=:\rho_{RA}\to \rho_R\otimes \rho_A$, (2) Quenching: The Hamiltonian is instantaneously changed from $\widehat{H}_A$ to $\beta^{-1}\ln \rho_A$ such that $\rho_A$ is an equilibrium state. (3) Isothermal quasistatic transformation of the Hamiltonian back to $\widehat{H}_A$. The extractable work from each step of the process is written in blue color. The thermodynamic utility of the channel is quantified as the maximal extractable work via the partial thermalization process, see Theorem \ref{['thm:excess']}.

Theorems & Definitions (54)

• Definition 1: Generalized free energy
• Proposition 1
• proof
• Theorem 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3: LMB25
• Proposition 2
• Lemma 4