For cyclic quantum heat engines coupled to lattice baths with short-range interactions, Shiraishi and Tajima (Phys. Rev. E 96, 022138 (2017)) rigorously proved that the efficiency η\eta satisfies η≤ηC−b/(2vLRτ+C)D\eta \le \eta_C - b/(2v_{\rm LR}\tau + C)^D, where ηC\eta_C is the Carnot efficiency, τ\tau is the cycle time, DD the spatial dimension, vLRv_{\rm LR} the Lieb-Robinson velocity, CC a correction term depending on the details of the bath and the engine, irrelevant for large τ\tau, and b∝1/jmax⁡b \propto 1/j_{\max} involves the volume-normalized Fisher information jmax⁡j_{\max} (a local heat-capacity-like quantity) of the cold bath. This excludes Carnot efficiency at finite power, but only for lattice baths with finite local susceptibility. The source paper itself explicitly leaves open whether the trade-off persists (i) for non-lattice, continuum baths; and its stated validity condition — jmax⁡j_{\max} is finite if the heat capacity of the bath is finite, so in particular if the bath is not at a first-order phase transition point — leaves unsettled (ii) whether a nontrivial trade-off persists for baths at or near first-order phase transitions or critical points, where jmax⁡j_{\max} diverges and the present bound degenerates to the bare Carnot limit. The problem is to settle the domain of validity of this no-go principle: either extend the bound rigorously to these regimes (possibly with modified time-dependence) or construct an explicit engine within them that attains Carnot efficiency at finite power.

Background

A heat engine converts heat from a hot reservoir into work while rejecting part of it to a cold reservoir; its efficiency η=W/QH\eta = W/Q^{\rm H} (work output per heat absorbed) is bounded by the Carnot efficiency ηC=1−TL/TH\eta_C = 1 - T_{\rm L}/T_{\rm H}. Whether an engine can operate at ηC\eta_C while delivering finite power (finite work per unit time, requiring cycle time τ<∞\tau < \infty) is a long-standing open problem of finite-time thermodynamics: the second law alone does not forbid it, since Carnot efficiency can be approached in quasi-static cycles whose power tends to zero. For classical Markovian engines — those whose working medium follows a memoryless stochastic dynamics with local detailed balance — Shiraishi and Saito (arXiv:1602.03645) rigorously proved that Carnot efficiency implies zero power. The quantum, non-Markovian case is harder: the engine may evolve unitarily together with its baths, and no Markovian master equation is available. Shiraishi and Tajima (arXiv:1701.01914) addressed it for quantum spin or fermion systems on a discrete lattice with short-range interactions. Their proof combines two ingredients. First, the Lieb-Robinson (LR) bound — a theorem stating that in quantum lattice systems with local, finite-strength interactions, operator spreading is confined to an effective light cone of finite velocity vLRv_{\rm LR}, the nonrelativistic counterpart of a speed of light — implies that after a finite-time cycle, most of the energy dumped into the cold bath still lies in a finite region near the engine, so the bath is locally out of equilibrium and entropy is produced. Second, quantum information geometry — specifically the quantum Fisher information associated with the Bogoliubov-Kubo-Mori inner product, which for thermal states reduces to the heat capacity — bounds from below the entropy production needed to re-thermalize that region; the relevant coefficient is jmax⁡j_{\max}, the Fisher information of a bath region normalized by its volume, which is finite whenever the bath's heat capacity per volume is finite. Combining the two yields η≤ηC−(QL)2/(8βLQHjmax⁡(2vLRτ+C)D)\eta \le \eta_C - (Q^{\rm L})^2/(8\beta^{\rm L} Q^{\rm H} j_{\max} (2v_{\rm LR}\tau + C)^D), i.e. a finite deficit from Carnot efficiency at any finite cycle time. The theorem's assumptions delimit its scope in two ways that the authors themselves flag. (i) Lattice structure: the LR bound is invoked for lattice systems; the paper states that if one succeeds in proving an appropriate LR-type bound for non-lattice (continuum or other) systems, the efficiency bound extends directly — but constructing that extension remains open. Most physical baths (electron gases, photon and phonon fields, trapped continua) are not lattices, and for them the mathematical status of locality bounds is only partially settled: operator-norm LR bounds are known for continuum many-fermion systems with ultraviolet-regularized interactions (Gebert, Nachtergaele, Reschke, Sims, arXiv:1912.12552; improved by Hinrichs, Lemm, Siebert, arXiv:2310.17736, and Bachmann and De Nittis, arXiv:2406.15670), while for interacting bosons — whose local Hilbert spaces are infinite-dimensional and local energies unbounded — only ensemble-averaged finite-speed results at finite density exist (Yin and Lucas, arXiv:2106.09726), with genuine operator-norm bounds explicitly described as notoriously difficult. (ii) Finite susceptibility: the coefficient jmax⁡j_{\max} is finite only if the bath's heat capacity per volume is finite; at or near a first-order phase transition (where latent heat makes the per-volume heat capacity jump or diverge) or near a continuous critical point (where it diverges with a critical exponent), jmax⁡j_{\max} may be infinite or ill-defined, the deficit coefficient vanishes, and the inequality degenerates to the Carnot bound. Whether a nontrivial trade-off survives in such singular-bath regimes, and in what form, is not addressed by the source paper. Related later work clarifies the surrounding landscape but does not touch these two gaps: the Markovian classical no-go (Shiraishi and Saito, arXiv:1602.03645); speed limits for open systems coupled to general environments, which bound state-transformation speed rather than engine efficiency (Shiraishi and Saito, arXiv:2002.11991); an exactly solvable two-terminal engine showing that the linear-response 'dream engine' with asymmetric Onsager coefficients (the linear-response transport coefficients relating thermodynamic currents to their driving forces) is destabilized (Lee, Park, Chun, Um and Park, Phys. Rev. E 101, 052132 (2020)); a classical underdamped Brownian Carnot cycle claimed to reconcile Carnot efficiency with finite power in a small-temperature-difference limit (Miura, Izumida and Okuda, arXiv:2011.12545); a coherence-based quantum cycle that effectively attains Carnot efficiency at finite power by evading the classical current-dissipation trade-off (Tajima and Funo, arXiv:2004.13412); a proposed superconducting-circuit engine with collectively enhanced dissipation approaching Carnot efficiency at finite power (Toma, Noguchi, Funo and Tajima, arXiv:2607.08713); classical stochastic Ising-cycle engines exploring working-medium criticality (Forão, Datta, Fiore and Barato, arXiv:2604.04730); and a unified framework for power-efficiency trade-offs of low-dissipation-type cycles (cycles whose irreversible entropy production scales inversely with cycle time; Ma and Fu, arXiv:2411.03849). None of these extends the Lieb-Robinson-based, non-Markovian efficiency bound beyond lattice baths or analyzes bath singularities in that setting; the same authors' later technique transfers the locality-plus-information-geometry idea to measurement and computation speed limits (Nakajima and Tajima, arXiv:2405.15291), again on lattices.

Problem Statement

Determine whether the rigorous efficiency-versus-time trade-off for cyclic quantum heat engines proven by Shiraishi and Tajima — η≤ηC−b/(2vLRτ+C)D\eta \le \eta_C - b/(2v_{\rm LR}\tau + C)^D with finite positive coefficients — extends beyond lattice baths with finite per-volume heat capacity, or fails there. The single scientific objective is to settle the domain of validity of the no-go principle that a finite-power cyclic quantum heat engine cannot attain (or asymptotically approach) the Carnot efficiency. Both regimes (a) and (b) below must be addressed for a complete resolution of this objective; each is independently refutable, and a result on either one alone is a partial contribution that leaves the other open. Two linked regimes, traceable to limitations stated in the source paper, define the target: (a) Non-lattice baths: for engines whose hot and cold reservoirs are continuum (or otherwise non-lattice) systems with short-range interactions, determine whether the finite-time efficiency bound extends to such baths. The source paper leaves this regime open, and no extension of the efficiency-versus-time bound to general continuum baths has since been constructed. (b) Thermodynamically singular baths: for baths at or near a first-order phase transition or critical point, where the volume-normalized Fisher information jmax⁡j_{\max} diverges and the proven bound degenerates to the bare Carnot inequality, determine whether a nontrivial trade-off between efficiency and cycle time survives — e.g. in a modified form with the τ−D\tau^{-D} scaling replaced by scaling involving the bath's critical behavior — or whether such singular baths permit explicit engine constructions with finite power and asymptotically Carnot efficiency.

References

  1. Naoto Shiraishi and Hiroyasu Tajima, 'Efficiency versus Speed in Quantum Heat Engines: Rigorous Constraint from Lieb-Robinson Bound', Phys. Rev. E 96, 022138 (2017). DOI: 10.1103/PhysRevE.96.022138, https://arxiv.org/abs/1701.01914 (source paper)
  2. Naoto Shiraishi and Keiji Saito, 'Incompatibility between Carnot efficiency and finite power in Markovian dynamics', arXiv:1602.03645 (2016), https://arxiv.org/abs/1602.03645
  3. Martin Gebert, Bruno Nachtergaele, Jake Reschke and Robert Sims, 'Lieb-Robinson bounds and strongly continuous dynamics for a class of many-body fermion systems in R^d', arXiv:1912.12552 (2019; published in Ann. Henri Poincare 21, 3609 (2020)). DOI: 10.1007/s00023-020-00959-5, https://arxiv.org/abs/1912.12552
  4. Benjamin Hinrichs, Marius Lemm and Oliver Siebert, 'On Lieb-Robinson bounds for a class of continuum fermions', arXiv:2310.17736 (2023), https://arxiv.org/abs/2310.17736
  5. Sven Bachmann and Giuseppe De Nittis, 'Lieb-Robinson bounds in the continuum via localized frames', arXiv:2406.15670 (2024), https://arxiv.org/abs/2406.15670
  6. Chao Yin and Andrew Lucas, 'Finite speed of quantum information in models of interacting bosons at finite density', arXiv:2106.09726 (2021; published as Phys. Rev. X 12, 021039 (2022)). DOI: 10.1103/PhysRevX.12.021039, https://arxiv.org/abs/2106.09726
  7. Naoto Shiraishi and Keiji Saito, 'Speed limit for open systems coupled to general environments', arXiv:2002.11991 (2020), https://arxiv.org/abs/2002.11991
  8. Jae Sung Lee, Jong-Min Park, Hyun-Myung Chun, Jaegon Um and Hyunggyu Park, 'Exactly solvable two-terminal heat engine with asymmetric Onsager coefficients: Origin of the power-efficiency bound', Phys. Rev. E 101, 052132 (2020). DOI: 10.1103/PhysRevE.101.052132, arXiv:2003.01869, https://arxiv.org/abs/2003.01869
  9. Kosuke Miura, Yuki Izumida and Koji Okuda, 'Compatibility of Carnot efficiency with finite power in an underdamped Brownian Carnot cycle in small temperature-difference regime', arXiv:2011.12545 (2020; published as Phys. Rev. E 103, 042125 (2021)). DOI: 10.1103/PhysRevE.103.042125, https://arxiv.org/abs/2011.12545
  10. Hiroyasu Tajima and Ken Funo, 'Superconducting-like heat current: Effective cancellation of current-dissipation trade off by quantum coherence', arXiv:2004.13412 (2020), https://arxiv.org/abs/2004.13412
  11. Shogo Toma, Atsushi Noguchi, Ken Funo and Hiroyasu Tajima, 'Approaching Carnot Efficiency at Finite Power in an Experimentally Feasible Quantum Heat Engine', arXiv:2607.08713 (2026), https://arxiv.org/abs/2607.08713
  12. Gustavo A. L. Forão, Arya Datta, Carlos E. Fiore and Andre C. Barato, 'Cyclic Heat Engine with the Ising model: role of interactions and criticality', arXiv:2604.04730 (2026; published as J. Stat. Mech. (2026) 083205). DOI: 10.1088/1742-5468/ae8ce8, https://arxiv.org/abs/2604.04730
  13. Yu-Han Ma and Cong Fu, 'Unified approach to power-efficiency trade-off relations of generic thermal machines', arXiv:2411.03849 (2024), https://arxiv.org/abs/2411.03849
  14. Satoshi Nakajima and Hiroyasu Tajima, 'Speed-Accuracy Trade-Off Relations in Quantum Measurements and Computations', arXiv:2405.15291 (2024), https://arxiv.org/abs/2405.15291
  15. Naoto Shiraishi, 'Stationary engines in and beyond the linear response regime at the Carnot efficiency', arXiv:1703.03621 (2017), https://arxiv.org/abs/1703.03621