Thermodynamic uncertainty relations (TURs) bound the precision of a current by the entropy production it costs, and they imply that steady-state heat engines face a strict trade-off between efficiency, power, and constancy (low power fluctuations). Miller, Mohammady, Perarnau-Llobet, and Guarnieri [Phys. Rev. Lett. 126, 210603 (2021)] established the first quantum TUR for periodically driven heat engines (PDHEs), deriving an efficiency bound with an explicit quantum-friction (skew-information) correction — but only in the slow-driving, weak-coupling, Markovian regime, closing with the statement that a full understanding of the role of quantum effects in PDHEs beyond slow driving and weak coupling remains open. Later work has advanced each leg — work statistics at arbitrarily fast (perturbative) driving, precision limits at arbitrary coupling, measurement-scheme-dependent quantum bounds, and trapped-ion measurements of quantum-friction corrections — yet no TUR-type efficiency–power–constancy bound valid at finite driving speed, beyond weak coupling and, in its strongest form, beyond Markovian dynamics, exists. The problem is to derive such a bound (or prove none of this form survives), with a measurement-scheme-consistent definition of the work statistics and an identified quantum correction.

Background

A thermodynamic uncertainty relation (TUR) is an inequality stating that the relative fluctuations (noise-to-signal ratio) of any nonequilibrium current cannot be made small without paying entropy production: for steady-state classical machines, Var(J)/⟨J⟩2≥2/Σ\mathrm{Var}(J)/\langle J\rangle^2 \geq 2/\Sigma, where Σ\Sigma is the entropy produced. For steady-state heat engines (SSHEs) — engines whose working medium sits in a stationary nonequilibrium state between two reservoirs — this yields a universal efficiency–power–constancy trade-off: an engine delivering finite average power with small power fluctuations cannot simultaneously approach the Carnot efficiency. Periodically driven heat engines (PDHEs) instead cycle a working medium by periodically modulating its Hamiltonian and the bath temperatures. Miller, Mohammady, Perarnau-Llobet, and Guarnieri [Phys. Rev. Lett. 126, 210603 (2021)] showed that PDHEs obey a different, weaker TUR: in the slow-driving limit, where the cycle duration τ\tau is long compared to the intrinsic equilibration time teqt^{eq} of the working medium, a PDHE can produce finite power with small fluctuations while approaching Carnot efficiency, the bound carrying a finite-time correction to Carnot at leading order in the driving speed. Their setting assumes weak system–bath coupling and Markovian dynamics governed by a Lindblad master equation (the standard Markovian quantum master equation for open systems) obeying detailed balance; work statistics are defined through quantum-jump monitoring of the system, and the paper notes that fluctuating work is predominant and depends on the measurement scheme used. Because the driving Hamiltonian at different times generally does not commute, the cycle generates coherences that are dissipated as heat — quantum friction — and the bound acquires an explicit quantum correction term built from the Wigner–Yanase skew information (a standard measure of the coherence between the energy eigenbasis of the instantaneous Hamiltonian and the actual state). This quantum term ηQ\eta^{Q} strictly reduces the achievable efficiency relative to power and reliability whenever the driving is non-commuting, establishing that quantum friction limits efficiency in this regime. Prior to and alongside this result, coherence was known to allow violations of the classical TUR in steady-state quantum systems (Menczel, Loisa, Brandner, Flindt, J. Phys. A 54, 314002 (2021)), so the fate of precision–dissipation trade-offs in the quantum regime is genuinely subtle. The 2021 result, however, is a leading-order-in-driving-speed statement: the derivation expands the full counting statistics (the generating function of work and heat fluctuations) in teq/τ≪1t^{eq}/\tau \ll 1, and nothing is known about the trade-off once this expansion fails — at finite or fast driving — or once weak coupling and Markovianity are relaxed. The authors explicitly leave 'a full understanding of the role of quantum effects in PDHEs beyond the slow-driving and weak-coupling regime' as an open question.

Problem Statement

Establish the thermodynamic uncertainty relation, and the associated efficiency–power–constancy trade-off, for cyclically (periodically) driven quantum heat engines outside the regime where it is currently proven — that is, valid at arbitrary finite cycle duration τ\tau comparable to or shorter than the working medium's equilibration time teqt^{eq} (finite or fast driving) and beyond weak system–bath coupling, in its strongest form also beyond Markovian Lindblad dynamics. Establish such a bound for a stated class of dynamics extending beyond the slow-driving regime and beyond weak system–bath coupling (in its strongest form, also beyond Markovian dynamics): define the work and heat statistics under a specified measurement scheme (quantum-jump monitoring, diffusive monitoring, or two-point energy measurements — projective measurements of the working medium's energy only at the start and end of the cycle), state the bound in a way that is either scheme-independent or explicit about its scheme dependence, identify the quantum correction term — the finite-speed analogue of the skew-information/quantum-friction correction of the slow-driving bound ηQ\eta^{Q} — and determine whether quantum coherence at finite driving speed can only degrade the efficiency–constancy trade-off, as it does under slow driving, or can enhance it beyond the classical bound, as coherence is known to do in some steady-state contexts; or demonstrate, via an explicit solvable model, that no universal entropy-production-based bound of this form holds at finite driving speed.

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