Resource-resolved quantum fluctuation theorems in end-point measurement scheme

TL;DR

This work develops a resource-resolved framework for quantum fluctuation theorems under end-point measurement, enabling the explicit isolation of athermality, coherence, and entanglement effects on nonequilibrium energy statistics. By replacing unphysical coherence operators with state-based, operational decompositions (athermality and coherence weights; BSA for entanglement), it yields four families of fluctuation theorems with clear resource-specific corrections to Jarzynski-type and Crooks-type relations. It also introduces practical measures—the coherence fluctuation distance and entanglement fluctuation distance—based on KL divergences between trajectory distributions, linking quantum resources to thermodynamic relevance in a process-dependent way. The framework is designed to be experimentally accessible and extensible to feedback control and multipartite networks, potentially guiding resource-aware thermodynamics in quantum technologies.

Abstract

Fluctuation theorems provide universal constraints on nonequilibrium energy and entropy fluctuations, making them a natural framework to assess how and to what extent quantum resources become thermodynamically relevant. We develop a unified framework for incorporating a generic quantum resource, including athermality, quantum coherence, and entanglement, into fluctuation theorems. We work within the end point measurement scheme, which avoids an initial energy measurement and allows quantum resources in the initial state to affect nonequilibrium energy statistics. We derive a family of quantum fluctuation theorems, including generalized Jarzynski equalities and Crooks type fluctuation relations, in which corrections decompose into resource resolved contributions. For single systems, we introduce the concept of weight of athermality, and combine it with the weight of coherence to isolate distinct thermodynamic effects of these quantum resources. For bipartite systems, we furthermore obtain two families of entanglement-resolved fluctuation theorems using an appended correlation operator and the best separable approximation, respectively. Finally, we introduce the concepts of coherence and entanglement fluctuation distances, as Kullback Leibler divergences, which quantify the thermodynamic relevance of quantum resources in a process-dependent and operational manner.

Figures (3)

• Figure 1: Zoo of fluctuation theorems. The figure provides a schematic classification of all fluctuation theorems (FTs) derived in this work. We organize the results according to the system structure: one-party (single) versus two-party (bipartite) quantum systems and the manner in which the initial resourceful quantum state is decomposed. For one-party scenarios, we distinguish between FTs obtained using the previously employed coherence-operator decomposition (orange pentagon), which leads to EPM Jarzynski equality and entropy-production relations derived in Refs. PhysRevA.104.L050203Hernandez-Gomez:2022xor, and a resource-theoretic decomposition (pink boxes) based on the weight of athermality (Def. \ref{['def:athermality']}) and weight of coherence (Def. \ref{['def:coherence']}). The latter yields refined FTs in which classical uncertainty, athermality, and coherence contributions are cleanly separated (Eq. \ref{['eq:Jarzynski-coh']}, Eq. \ref{['eq:DFT-entropy-single-party']} and Eq. \ref{['eq:Theta-Sigma-Final']}). For two-party case, we first consider a correlation-operator decomposition (orange pentagon), which captures total correlations leading to correlation-corrected FTs (Eq. \ref{['eq:Jarzynski-EnAB-biparty']} and Eq. \ref{['eq:DFT-entropy-bi-party']} with Eq. \ref{['eq:DeltaPsi-residual']}). We then introduce a fully operational and resource-theoretic formulation based on the Best Separable Approximation (Def. \ref{['def:BSA']}), which yields entanglement-resolved FTs (Eq. \ref{['eq:Jarzynski-ent']} and Eq. \ref{['eq:DFT-entropy-bi-party']} with Eq. \ref{['eq:Corre-corrections']}). Together, the four branches shown in the diagram constitute a unified framework for systematically isolating and quantifying the thermodynamic roles of coherence, athermality, and entanglement in non-equilibrium quantum fluctuation relations within the EPM protocol.
• Figure 2: Coherence fluctuation distance and its upper bounds for single-qubit initial states: unitary dynamics.The single qubit initial state is given by Eq. \ref{['eq:ini-coh']}, parametrized by coherence strength $\gamma$. The coherence fluctuation distance $\mathfrak{D}_c (\rho_i)$ (solid red) is plotted as a function of $\gamma$ together with its two upper bounds from Eq. \ref{['eq:CFD-bounds']}: the intermediate bound (blue dashed), and the protocol-independent maximal bound $2 C_{re}(\rho_i)$ (brown dot–dashed). The time evolution is unitary under the time-dependent Hamiltonian of Eq. \ref{['eq:Ham-CFD-Unitary']}. Parameters values used are $t_i = 0$, $t_f = 10$, $a = 0.9$, $\Omega=1$, $\omega=1$.
• Figure 3: Coherence fluctuation distance and its upper bounds for single-qubit initial states: dissipative dynamics.The single qubit initial state is given by Eq. \ref{['eq:ini-coh']}, parametrized by coherence strength $\gamma$. The coherence fluctuation distance $\mathfrak{D}_c (\rho_i)$ (solid red) is plotted as a function of $\gamma$ together with its two upper bounds from Eq. \ref{['eq:CFD-bounds']}: the intermediate bound (blue dashed), and the protocol-independent maximal bound $2 C_{re}(\rho_i)$ (brown dot–dashed). The time evolution is governed by Lindblad master equation, Eq. \ref{['eq:master-eqn']}, with the time-dependent Hamiltonian given by Eq. \ref{['eq:Ham-CFD-Unitary']} and Linblad operator $L = \sigma_x$. Parameters values used are $t_i = 0$, $t_f = 10$, $a = 0.9$, $\Omega=1$, $\omega=1$, $\kappa = 0.1$.

Theorems & Definitions (10)

• Definition 1: Weight of athermality
• Definition 2: Weight of coherence PhysRevA.97.032342PhysRevA.102.032406
• Definition 3: Best Separable Approximation PhysRevLett.80.2261
• Definition 4: Phase-covariant qubit dynamics Filippov:2019zdz
• Theorem 5
• proof
• Lemma 6
• proof
• proof
• proof