Rubino, Hovhannisyan and Skrzypczyk (npj Quantum Information 11, 102 (2025)) proved that any state-independent quantum work-measurement scheme that certifies energy conservation, i.e. satisfies ∑aWaMa=Ω\sum_a W_a M_a = \Omega with Ω=U†H′U−H\Omega = U^\dagger H' U - H the Heisenberg operator of work, obeys a quantum-corrected Jarzynski equality ⟨e−βW⟩S=e−βΔF+ΞS\langle e^{-\beta W}\rangle_{\mathcal S} = e^{-\beta\Delta F + \Xi_{\mathcal S}} with a strictly positive coherence-induced correction ΞS\Xi_{\mathcal S}. They established a closed-form upper bound via the Heisenberg-work-operator (HOW) scheme, proved by counterexample that this bound is generally not attained, and constructed modified two-point-measurement schemes: a first modification TPMϵ\mathrm{TPM}_\epsilon whose correction necessarily diverges as O(βw/ϵ)O(\beta w/\epsilon) when ϵ→0\epsilon \to 0, and a pragmatic scheme TPMϵ,V\mathrm{TPM}_{\epsilon,V} whose correction stays O(1)O(1) as ϵ→0\epsilon \to 0. The open problem is the completion of this framework: (i) the general characterization and minimization of the correction Ξ\Xi over energy-conserving schemes (no closed-form expression for the minimum is known, and numerics suggest optimal schemes are exotic); (ii) extension to infinite-dimensional systems; (iii) extension to the non-unitary dynamics of resource-theoretic work-storage (weight) systems; and (iv) removal of the schemes' dependence on prior knowledge of HH, H′H', UU.

Background

In thermodynamics at the nanoscale, work is a random quantity fluctuating between repetitions of a driving protocol, and fluctuation relations constrain its statistics. The Jarzynski equality (JE) states that when a system starts in thermal equilibrium at inverse temperature β\beta — in the Gibbs state τβ=e−βH/Z\tau_\beta = e^{-\beta H}/Z with partition function ZZ — the exponential average of the work WW over a protocol arbitrarily far from equilibrium equals e−βΔFe^{-\beta\Delta F}, where ΔF\Delta F is the equilibrium free-energy difference between the initial Hamiltonian HH and the final Hamiltonian H′H'. In quantum mechanics, work is not an observable of a single state but a quantity assigned to a driven process, so any operational definition must specify a measurement scheme and the disturbance it induces. The standard choice is the two-point measurement (TPM) protocol: projectively measure the energy at the start, which collapses the state onto an energy eigenstate; let the system evolve under the driving unitary UU; projectively measure the energy at the end; and define work as the difference of the two outcomes. TPM satisfies the JE exactly for thermal initial states, but its first measurement destroys superpositions of energy eigenstates (coherence), so for coherent initial states the measured average work differs from the true average energy change Tr(H′UρU†)−Tr(Hρ)\mathrm{Tr}(H' U \rho U^\dagger) - \mathrm{Tr}(H\rho) of the unmeasured process: TPM fails to certify energy conservation.

A general measurement scheme is specified by a positive-operator-valued measure (POVM) — a set of positive operators Ma≥0M_a \ge 0 with ∑aMa=1\sum_a M_a = \mathbb{1} — together with an assignment of a work value WaW_a to each outcome aa; it is state-independent if neither the POVM nor the outcomes depend on the initial state ρ\rho. Perarnau-Llobet, Bäumer, Hovhannisyan, Huber and Acín (2017) proved a no-go theorem: no state-independent scheme can simultaneously reproduce TPM statistics on incoherent states and the correct average work on coherent states. Hovhannisyan and Imparato (2024) sharpened the conflict: demanding energy conservation — the measured average work equals the true energy change for every initial state, which is equivalent to the operator constraint ∑aWaMa=Ω\sum_a W_a M_a = \Omega with Ω=U†H′U−H\Omega = U^\dagger H' U - H the Heisenberg operator of work (HOW) — is incompatible with the JE whenever [U†H′U,H]≠0[U^\dagger H' U, H] \ne 0.

Rubino, Hovhannisyan and Skrzypczyk (2025) resolved the tension in one direction: keep energy conservation and accept a modified fluctuation relation. They proved that any energy-conserving state-independent scheme obeys a quantum-corrected Jarzynski equality ⟨e−βW⟩S=e−βΔF+ΞS\langle e^{-\beta W}\rangle_{\mathcal S} = e^{-\beta\Delta F + \Xi_{\mathcal S}}, where the correction ΞS≥0\Xi_{\mathcal S} \ge 0 is strictly positive exactly in the coherent case [U†H′U,H]≠0[U^\dagger H' U, H] \ne 0; consequently, apparent violations of the second law become more probable than classically, with the tail bound Prob(W−ΔF≤−ζ)≤eΞS−βζ\mathrm{Prob}(W - \Delta F \le -\zeta) \le e^{\Xi_{\mathcal S} - \beta\zeta}. Defining the minimal correction Ξ\Xi as the minimum of ΞS\Xi_{\mathcal S} over all energy-conserving schemes, they showed that the projective measurement of Ω\Omega (the HOW scheme) yields the closed-form upper bound Ξ≤ΞHOW=βΔF+ln⁡ Tr[τβe−βΩ]\Xi \le \Xi_{\mathrm{HOW}} = \beta\Delta F + \ln\,\mathrm{Tr}[\tau_\beta e^{-\beta\Omega}], constructed an explicit counterexample proving that the HOW scheme is not generally optimal, and constructed modified TPM schemes: a first modification TPMϵ\mathrm{TPM}_\epsilon, whose outlier outcomes must scale as w/ϵw/\epsilon (with ww a natural work scale, e.g. w=Tr[H2]+Tr[(H′)2]w = \sqrt{\mathrm{Tr}[H^2] + \mathrm{Tr}[(H')^2]}) so that its correction diverges as ΞTPMϵ=O(βw/ϵ)\Xi_{\mathrm{TPM}_\epsilon} = O(\beta w/\epsilon) when ϵ→0\epsilon \to 0 — an exponential violation of the Jarzynski equality — and a pragmatic scheme TPMϵ,V\mathrm{TPM}_{\epsilon,V}, so named because its outcome values are shifted by an amount VwVw from the TPM values, whose correction remains O(1)O(1) as ϵ→0\epsilon \to 0 while the scheme stays close to TPM; both come with explicit quantum-circuit implementations. The analysis is restricted to finite-dimensional systems with discrete outcomes, unitary driving, and schemes that presuppose knowledge of HH, H′H', and UU.

Problem Statement

The problem is to complete the energy-conserving quantum work-fluctuation framework of Rubino, Hovhannisyan and Skrzypczyk (2025). Its common objective is to determine exactly how much the Jarzynski equality must change when quantum work measurement is required to certify energy conservation, and in which physical settings that corrected relation can be enforced. Four linked questions, all posed as open by the source:

(1) Core — the minimal quantum correction. For fixed finite-dimensional HH, H′H' and driving unitary UU, characterize

Ξ  =  min⁡{Ma,Wa}: ∑aWaMa=Ω ΞS,ΞS=βΔF+ln⁡⟨e−βW⟩S,\Xi \;=\; \min_{\{M_a, W_a\}:\ \sum_a W_a M_a = \Omega}\ \Xi_{\mathcal S}, \qquad \Xi_{\mathcal S} = \beta\Delta F + \ln \langle e^{-\beta W}\rangle_{\mathcal S},

where Ω=U†H′U−H\Omega = U^\dagger H' U - H: determine the optimizing scheme or schemes (the POVM elements and outcome assignments attaining the minimum), decide whether Ξ\Xi admits a closed-form expression in HH, H′H', UU and β\beta (defined operationally as an expression obtainable by finitely many applications of matrix exponential and logarithm, matrix inverse, trace, and arithmetic operations), and settle tightness — for which processes the HOW scheme (or any simple TPM-like scheme) attains the minimum, and how far below the closed-form upper bound ΞHOW=βΔF+ln⁡ Tr[τβe−βΩ]\Xi_{\mathrm{HOW}} = \beta\Delta F + \ln\,\mathrm{Tr}[\tau_\beta e^{-\beta\Omega}] the true minimum lies, given the source's counterexample proving the bound is generally not attained.

(2) Infinite dimensions. Extend the corrected Jarzynski equality and its energy-conserving measurement schemes to systems with infinite-dimensional Hilbert spaces (and continuous outcome spectra), beyond the finite-dimensional, discrete-outcome setting of the source; it is required to capture the corrected Jarzynski equality and a class of energy-conserving schemes with continuous outcome spectra, and to state and support whether the optimizer, closed-form, and tightness conclusions of (1) carry over.

(3) Non-unitary dynamics. Extend the framework to the non-unitary evolution of the work-storage ("weight") system of resource-theoretic quantum thermodynamics, where work is the energy change of an explicitly modeled weight that generally starts in a coherent state and evolves non-unitarily; it is required to capture a corrected fluctuation relation for this setting, to construct energy-conserving measurement schemes for the weight's energy change, and to state whether the correction remains non-negative (and strictly positive in the coherent case) as in the unitary framework.

(4) Knowledge independence. Remove or quantify the schemes' reliance on prior knowledge of the system Hamiltonians and evolution, i.e. construct energy-conserving work measurements that operate without full characterization of HH, H′H' and UU. Quantifying the residual reliance — stating explicitly which prior knowledge is indispensable and which can be dropped — resolves (4); complete removal of the dependence is the stronger target and need not be attained.

All four questions are mandatory: the task is the completion of the framework, so a proper subset — the core question (1) alone included — is not a full resolution. Within (1), the optimizing scheme, the existence or impossibility of a closed form for Ξ\Xi, and the tightness of ΞHOW\Xi_{\mathrm{HOW}} relative to the true minimum are each required, the last expected to be settled as a consequence of the other two.

These components jointly support one objective: an operational theory of energy-conserving quantum work measurement whose fluctuation-relation modification is quantitatively minimal and whose domain covers the settings — infinite-dimensional systems, resource-theoretic work storage, imperfectly characterized processes — where work measurement is actually required. Experimentally practical, platform-tailored circuit realizations of the corrected schemes are the applied face of the same objective and are not a separate deliverable.

References

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