In the energy-flux formalism for autonomous bipartite quantum systems introduced by Weimer et al. and developed by Hossein-Nejad, O'Reilly and Olaya-Castro [New J. Phys. 17, 075014 (2015)], the energy change of each subsystem is partitioned into heat and work by reference to a local effective measurement basis (LEMBAS): energy changes that alter the local von Neumann entropy are heat, those that do not are work. The formalism yields closed-form fluxes for each commutation class of the interaction, extends to partitions coupled to Markovian thermal baths, and is compatible with the second law through an exact entropy decomposition. Two acknowledged issues remain open. First, the fluxes lack an operational measurement protocol: work is not a quantum observable, its relation to extractable work, and the backaction of the measurement apparatus on the very entropy that defines the partition, remain under scrutiny. Second, the practical applicability of the partitioning to entropy fluxes out of thermal equilibrium is unsettled: the Clausius-type relation dS=β dQdS=\beta\,dQ fails in general and the instantaneous entropy-production rate can be negative for non-Markovian reduced dynamics. The problem remains open: competing partitions (LEMBAS, entropy-based, ergotropy/complete-passivity-based, minimal-dissipation) still disagree on the same spin-oscillator models, and the most recent dedicated comparison (Seegebrecht and Schilling 2024) defers the question of which formulation is best suited for discussing accessibility in measurements.

Background

Quantum thermodynamics extends the laws of thermodynamics to systems whose size, coupling strength, or coherence lie outside the textbook regime of a macroscopic body weakly coupled to an ideal heat bath and driven by a classical external parameter. A central piece of this program is the split of a system's internal-energy change into work and heat. For a system with a time-dependent Hamiltonian H(t)H(t) driven by a classical field, the standard (Alicki-type) split assigns W˙=Tr{ρH˙}\dot W=\mathrm{Tr}\{\rho\dot H\} (energy injected by changing the Hamiltonian) and Q˙=Tr{ρ˙H}\dot Q=\mathrm{Tr}\{\dot\rho H\} (energy injected by changing the state). This presupposes an external classical drive. In an autonomous bipartite quantum system, with total Hamiltonian H=H1+H2+H12H=H_1+H_2+H_{12}, H12=A1⊗B2H_{12}=A_1\otimes B_2, and no external time dependence, both the 'drive' and the 'bath' are quantum systems of comparable size, and the split is no longer given.

Weimer, Henrich, Rempp, Schröder and Mahler proposed a basis-dependent answer, the energy-flux formalism. One introduces a local effective measurement basis (LEMBAS), the eigenbasis of the local energy operator H1′(t)=H1+H1,aeff(t)H_1'(t)=H_1+H_{1,a}^{\mathrm{eff}}(t), where the effective Hamiltonian H1eff(t)=A1⟨B2⟩tH_1^{\mathrm{eff}}(t)=A_1\langle B_2\rangle_t is the mean field exerted by the partner subsystem, split into a part H1,aeffH_{1,a}^{\mathrm{eff}} commuting with H1H_1 and a part H1,beffH_{1,b}^{\mathrm{eff}} that does not. Work is the energy change that leaves the local von Neumann entropy S1=−Tr{ρ1ln⁡ρ1}S_1=-\mathrm{Tr}\{\rho_1\ln\rho_1\} unchanged; heat is the energy change that alters it. Hossein-Nejad, O'Reilly and Olaya-Castro worked out the consequences in closed form for the distinct commutation classes of A1A_1 with H1H_1 and B2B_2 with H2H_2: fully commuting interactions carry no heat flux yet can dephase the reduced dynamics (entropy change without heat); partially commuting interactions admit heat only into the non-commuting side and require inter-partition correlations C12(t)=ρ(t)−ρ1(t)⊗ρ2(t)C_{12}(t)=\rho(t)-\rho_1(t)\otimes\rho_2(t); and the formalism extends to partitions coupled to Markovian thermal baths, whose action is a Lindblad dissipator L1\mathcal{L}_1 (the standard generator of Markovian open-system evolution) entering a time-local master equation ρ˙1=−i[H1,ρ1]+L1[ρ1]\dot\rho_1=-i[H_1,\rho_1]+\mathcal{L}_1[\rho_1], valid under the Born (weak coupling, near-factorized state) and Markov (rapid bath memory decay) approximations.

The same paper reconciled the formalism with the second law using the exact decomposition of Esposito, Lindenberg and Van den Broeck: for initially separable states evolving unitarily, ΔS1=ΔS1ir+ΔS1re\Delta S_1=\Delta S_1^{\mathrm{ir}}+\Delta S_1^{\mathrm{re}}, with irreversible part ΔS1ir=D[ρ(τ)∥ρ1(τ)ρ2(0)]≥0\Delta S_1^{\mathrm{ir}}=D[\rho(\tau)\|\rho_1(\tau)\rho_2(0)]\ge 0, where D[ρ∥σ]=Tr{ρln⁡ρ}−Tr{ρln⁡σ}D[\rho\|\sigma]=\mathrm{Tr}\{\rho\ln\rho\}-\mathrm{Tr}\{\rho\ln\sigma\} is the quantum relative entropy, and a reversible part that is not a heat flux. The direct Clausius-type identification dS1=β∗dQ1dS_1=\beta^*dQ_1 fails out of equilibrium, and although the integrated entropy production is nonnegative, its instantaneous rate can be negative whenever the reduced dynamics is non-Markovian (with memory, so that the reduced evolution is not a time-homogeneous quantum dynamical semigroup); positivity of the rate is recovered only in the Markov limit.

Two issues were explicitly left open by that work. First, the operational measurement of the fluxes: work is not a quantum observable, and the relation between the LEMBAS work flux and extractable, measurable work, together with the backaction of the measurement apparatus, 'remain under scrutiny'. Second, the practical applicability of the energy partitioning to the entropy fluxes of systems out of thermal equilibrium, where the heat-entropy correspondence that the formalism presupposes at equilibrium is known to fail in general.

Problem Statement

For an autonomous bipartite quantum system with Hamiltonian H=H1+H2+H12H=H_1+H_2+H_{12}, H12=A1⊗B2H_{12}=A_1\otimes B_2 (and its extension where one or both partitions couple to Markovian thermal baths), the energy-flux formalism partitions each subsystem's energy change into heat and work relative to a local effective measurement basis (LEMBAS): work fluxes are energy changes that leave the local von Neumann entropy unchanged, heat fluxes are energy changes that alter it. Establish the operational status of this partitioning, comprising two linked questions with a single common objective - deciding whether the LEMBAS work and heat fluxes are measurable physical quantities or a purely formal bookkeeping device:

(1) Measurement. Since work is not a quantum observable, construct an operational measurement protocol - or a proof of impossibility - that relates the LEMBAS work flux W˙1(t)\dot W_1(t) and heat flux Q˙1(t)\dot Q_1(t) to experimentally measurable quantities (for example extractable work or two-point energy-measurement statistics), including an accounting of the backaction of the measurement apparatus on the local entropy that defines the partition, and a constructive rule for identifying the local effective measurement basis set by the actual experimental apparatus that determines the LEMBAS.

(2) Entropy fluxes. Establish the practical applicability of the energy partitioning to entropy fluxes: determine the relation between the heat flux and the entropy change of a finite bipartite system out of thermal equilibrium - general enough to cover non-Markovian reduced dynamics, where the instantaneous entropy-production rate can transiently be negative while the integrated production stays nonnegative - under which the partition remains consistent with the second law and operationally meaningful.

A complete answer decides both questions for the formalism as stated, without restricting to a single platform, coupling class, or the Markovian limit. If (1) is resolved negatively, (2) reduces to exhibiting the second-law-consistent heat-entropy relation for the formal bookkeeping fluxes. Partial answers restricted to specific platforms or alternative partitions address derived subproblems but do not close the problem.

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