Finite-time quantum information engines — cyclic measurement-and-feedback machines whose work extraction is driven by information acquired during a measurement of finite duration — have had their power, efficiency, and information-to-work trade-off surfaces charted by Pareto optimisation, but under an idealization that removes the classical readout apparatus from the thermodynamic balance: the Landauer erasure cost of resetting the meter record is set to zero by assuming an arbitrarily cold reset device, and the time taken to read out and reset the meter is excluded from the cycle time. The open problem addressed here is the full thermodynamic accounting of finite-time quantum information engines: include the energetic (Landauer) and temporal costs of the classical readout and meter reset at finite reset temperature, and determine how the power–efficiency–information (and precision) Pareto surfaces move — and whether the identified optimal operating regimes of simultaneously positive net power and high efficiency survive.

Background

An information engine (or Maxwell-demon engine) is a cyclic machine that converts information about the state of a system into useful work: a measurement acquires information about the working medium, and feedback conditioned on the measurement outcome extracts work — in principle even from a single heat bath, because the generalized second laws of thermodynamics credit the acquired information as a thermodynamic resource. Quantum information engines implement this cycle with a quantum working medium, and variants have been realized in superconducting circuits, trapped ions, and nanomechanical systems.

In the finite-time quantum information engine of Hagman et al. (2025), the working medium S is a two-level system (TLS) with level spacing ΔE\Delta E, and the meter M is a quantum harmonic oscillator of frequency ω\omega, temporarily coupled to S during a measurement time tmt_m through an interaction that is switched on and off (a von-Neumann-type premeasurement, in which a quantum meter first becomes correlated with the system and is only afterwards collapsed by a classical apparatus). The coupling and decoupling cost a measurement work WmeasW_{\rm meas}, and the information transferred is quantified by the mutual information I(tm)I(t_m) shared between the system state and the meter outcome. The meter is then read out projectively by a classical apparatus, and work is extracted conditionally on the outcome via ergotropy — the maximum energy extractable from a quantum state by a cyclic unitary operation, realized for the TLS by a population-inverting π\pi-pulse — after which system and meter rethermalize with baths at temperatures TST_S and TMT_M. Performance is quantified by the output power Π=(⟨Wext⟩−Wmeas)/tm\Pi=(\langle W_{\rm ext}\rangle-W_{\rm meas})/t_m, i.e., the net extracted work (extracted work minus measurement work) divided by the cycle time, which is taken to be the measurement time tmt_m; by the thermodynamic efficiency ηHE=(⟨Wext⟩−Wmeas)/QS\eta_{\rm HE}=(\langle W_{\rm ext}\rangle-W_{\rm meas})/Q_S, comparing the net extracted work with the heat QSQ_S drawn from the system bath; and by the information-to-work conversion efficiency ηinfo=⟨Wext⟩/(TSI)\eta_{\rm info}=\langle W_{\rm ext}\rangle/(T_S I). Because these metrics are interdependent through tmt_m and the engine parameters, they are optimized simultaneously: Pareto optimality, a concept imported from economics and engineering, identifies the set of parameter choices — the Pareto front — at which no metric can be improved without worsening another. Hagman et al. computed these fronts for the TLS–oscillator engine, finding, among other results, a global maximum of power at a finite value of the efficiency at a specific finite measurement time, and distinct heat-engine versus heat-valve operating regimes (net work output versus purely heat-routing operation).

Two idealizations delimit that analysis. First, all metrics are cycle-averaged; at the nanoscale the power and efficiency fluctuate from cycle to cycle, and the noise-to-signal ratio of the output (the ratio of the variance of the extracted work to its squared mean; its inverse is called the precision) becomes a design consideration. Second, the classical apparatus that registers the meter outcomes is assumed to cost nothing: by Landauer's erasure principle, resetting a memory that has absorbed an entropy SrecS_{\rm rec} requires at least the work Wreset=TresetSrecW_{\rm reset}=T_{\rm reset}S_{\rm rec} (per erased bit, kBTln⁡2k_B T\ln 2), and the readout and erasure also take time. The analysis chooses the reset temperature arbitrarily small and keeps the reset outside the cycle-time bookkeeping, which removes both costs from the engine's thermodynamic balance. Subsequent work by two of the same authors (Berx, Olander and Kirchberg, 2026) resolved the first idealization for the same engine: it computed the Pareto front between mean extracted work and its fluctuations, showed that raising the precision costs information consumption, engine cycles, operation time, and average output, and obtained the work statistics and the front analytically in the cold-meter limit. That work retains the second idealization in exactly the same form, stating that the Landauer erasure cost of resetting the classical measurement device is negligible for a readout performed at arbitrarily low temperature. Further 2026 studies include measurement and erasure costs in related settings — daemonic (one-shot, measurement-assisted) work extraction with non-ideal quantum non-demolition energy measurements at finite auxiliary-bath temperature, and a trapped-ion measurement-feedback engine whose correlated outcome record lowers the reversible record-reset cost from the one-symbol entropy to the entropy rate (the asymptotic entropy per recorded symbol once correlations between successive outcomes are accounted for) — but none of them folds the readout-and-reset apparatus of a cyclic quantum information engine into its thermodynamic balance and re-derives the engine's optimal operating regimes.

Problem Statement

For finite-time quantum information engines of the general system–meter type introduced by Hagman et al. (2025) — cyclic measurement-and-feedback engines whose prototype implementation is a two-level system measured by a temporarily coupled quantum harmonic oscillator — include the classical readout-and-reset apparatus in the engine's thermodynamic balance, removing the idealization of an arbitrarily cold and un-timed reset. Concretely: (1) account for the energetic cost of erasing the classical meter record, Wreset=TresetSrecW_{\rm reset}=T_{\rm reset}S_{\rm rec} at a finite reset temperature TresetT_{\rm reset}, where SrecS_{\rm rec} is the entropy transferred from the quantum meter to the classical apparatus, together with any energetic cost of the readout interaction itself; and (2) account for the temporal cost, i.e., the finite durations of the classical readout and of the erasure protocol, inside the cycle time that defines the output power. With these costs included, characterize the trade-off surface among output power, thermodynamic efficiency, and information-to-work conversion efficiency — the Pareto fronts and the displacement of the previously identified optimal regimes (such as maximum power at finite efficiency at an optimal finite measurement time) — and determine the efficiency interval over which the net power remains positive, reporting its upper endpoint; this establishes whether regimes of simultaneously positive net power and high efficiency survive the full-cost accounting, where high efficiency means a thermodynamic efficiency above the value at maximum power on the reference (zero-cost) front. The energetic accounting, the temporal accounting, the Pareto-front characterization, and the survival determination are parts of this question taken together; the precision of the extracted work (the inverse noise-to-signal ratio) belongs to this question only in the fluctuation-resolved treatment. The problem is posed at the level of the general system–meter framework and is not restricted to the two-level-system/oscillator prototype; treatments of other meter models or working media are admissible realizations of the same question.

References

  1. R. Hagman, J. Berx, J. Splettstoesser, H. Kirchberg, 'Optimising finite-time quantum information engines using Pareto bounds', New Journal of Physics 27, 114507 (2025). DOI: 10.1088/1367-2630/ae18be; arXiv:2507.00712. https://arxiv.org/abs/2507.00712
  2. J. Berx, T. Olander, H. Kirchberg, 'Pareto-optimal work extraction and the thermodynamic cost of precision in quantum information engines' (2026). arXiv:2607.14973. https://arxiv.org/abs/2607.14973
  3. H. Kirchberg, A. Nitzan, 'Quantum information engines: Bounds on performance metrics by measurement time', Physical Review A 112, 032201 (2025). DOI: 10.1103/vdm1-4ryg; arXiv:2505.00686. https://arxiv.org/abs/2505.00686
  4. D. Morrone, F. Albarelli, V. Giovannetti, M. Paternostro, M. G. Genoni, 'Energetics in daemonic work extraction protocols via non-ideal QND-energy measurement' (2026). arXiv:2608.20487. https://arxiv.org/abs/2608.20487
  5. Y. Hong, D. Liu, J. Wei, L. Yan, J. Wang, 'Measurement-Feedback Quantum Information Engine: Coherence-Transition Interference and Correlated Work Statistics' (2026). arXiv:2608.14493. https://arxiv.org/abs/2608.14493
  6. J. Wei, Y. Hong, D. Liu, X. Wang, Z. Wang, Y. Zhan, K. Cui, J. Bu, J. Wang, L. Yan, G. Chen, 'Experimental Demonstration of a Measurement-Feedback Quantum Information Engine' (2026). arXiv:2607.28702. https://arxiv.org/abs/2607.28702
  7. B. Barman, J. Splettstoesser, H. Kirchberg, 'Activating thermally charged quantum batteries in finite time: Thermodynamic trade-offs between correlations, work, and information' (2026). arXiv:2607.12573. https://arxiv.org/abs/2607.12573
  8. P. A. Erdman, A. Rolandi, P. Abiuso, M. Perarnau-Llobet, F. Noé, 'Pareto-optimal cycles for power, efficiency and fluctuations of quantum heat engines using reinforcement learning', Physical Review Research 5, L022017 (2023). DOI: 10.1103/PhysRevResearch.5.L022017; arXiv:2207.13104. https://arxiv.org/abs/2207.13104
  9. G. A. L. Forão, J. Berx, C. E. Fiore, 'Characterization and optimization of heat engines: Pareto-optimal fronts and universal features', New Journal of Physics 27, 074605 (2025). DOI: 10.1088/1367-2630/adf08d; arXiv:2504.21717. https://arxiv.org/abs/2504.21717
  10. G. Paneru, S. Dutta, T. Sagawa, T. Tlusty, H. K. Pak, 'Efficiency fluctuations and noise induced refrigerator-to-heater transition in information engines', Nature Communications 11 (2020). DOI: 10.1038/s41467-020-14823-x. https://doi.org/10.1038/s41467-020-14823-x
  11. J. Monsel, M. Acciai, R. Sánchez, J. Splettstoesser, 'Autonomous demon exploiting heat and information at the trajectory level', Physical Review B 111, 045419 (2025). DOI: 10.1103/PhysRevB.111.045419; arXiv:2409.05823. https://arxiv.org/abs/2409.05823
  12. P. Fadler, A. Friedenberger, E. Lutz, 'Efficiency at Maximum Power of a Carnot Quantum Information Engine', Physical Review Letters 130, 240401 (2023). DOI: 10.1103/PhysRevLett.130.240401; arXiv:2301.13560. https://arxiv.org/abs/2301.13560
  13. H. Feng, J. Liu, 'Manipulation of information flow and thermodynamic performance in nonreciprocal quantum dot information engines' (2026). arXiv:2605.24554. https://arxiv.org/abs/2605.24554
  14. W. Chen, M. Chen, Y.-H. Ma, 'Finite-Time Thermodynamics of an Autonomous Information Machine' (2026). arXiv:2604.15953. https://arxiv.org/abs/2604.15953