Attaining Carnot efficiency with quantum and nanoscale heat engines

The paper addresses whether quantum and nanoscale heat engines operating in the one-shot finite-size regime can attain Carnot efficiency, despite inherent work fluctuations and irreversibility typical of small systems. Traditional thermodynamics assumes asymptotically large systems with vanishing fluctuations, whereas small quantum systems (N finite) exhibit significant fluctuations and limited deterministic work. Existing frameworks—fluctuation theorems and quantum information-theoretic resource theories—mostly consider single-bath interactions and do not provide a full resource-theoretic treatment of engines interacting with multiple baths. The authors propose a new operational framework—semi-local thermal operations (SLTOs)—where a bipartite working system interacts simultaneously with two baths, enabling a one-step engine cycle. They formulate a resource theory for such engines, derive many second laws governing state transformations, and show that reversible one-shot transformations are achievable when states are energy eigenstates, thereby enabling Carnot efficiency.

Two main approaches have extended thermodynamics to finite-size and quantum regimes: (i) fluctuation theorems (Jarzynski, Crooks, Campisi et al.) that connect nonequilibrium work statistics with free-energy differences; (ii) quantum information-theoretic resource theories (Brandão, Horodecki, Oppenheim, Wehner, Åberg, Lostaglio et al.) that formalize thermodynamic state transformations under thermal operations, defining second laws via Rényi divergences and addressing coherence constraints. Prior works often restrict to single-bath interactions, quantify single-shot work extraction limits, and highlight fundamental irreversibility and many second laws. Studies on nanoscale engines, finite-size effects, and anomalous phenomena (e.g., correlation-enabled heat flow, surpassing Carnot with imperfect work, dependence of efficiency beyond temperature) exist, but a comprehensive resource theory for engines with two baths in the one-shot regime and protocols achieving Carnot efficiency had been lacking. This work fills that gap by defining SLTOs/CSLTOs, semi-Gibbs states, and deriving necessary and sufficient second-law conditions for multi-bath engine transformations.

• Engine model: A bipartite working system S12 with non-interacting subsystems S1 and S2 (HS12 = HS1 + HS2) interacts semi-locally with two heat baths B1 and B2 at inverse temperatures β1 < β2. Operations are performed in a one-step cycle by swapping states and/or Hamiltonians of the subsystems, effectively compressing the traditional four-step Carnot cycle into a single stroke.
• Semi-local thermal operations (SLTOs): Free operations defined by a global unitary U acting on B1, B2, and S12, with two constraints: (1) exact total energy conservation [U, HB1+HS1+HB2+HS2] = 0, and (2) exact conservation of weighted energy [U, β1(HB1+HS1) + β2(HB2+HS2)] = 0. These jointly constitute a first law for engines in this regime. SLTOs reduce to standard local thermal operations when β1 = β2.
• Catalytic SLTOs (CSLTOs): Extension with a bipartite catalyst C12 = C1⊗C2 (HC = HC1 + HC2) that is returned unchanged. CSLTOs strictly map semi-Gibbs states to themselves and enlarge the set of allowed operations while preserving the first-law constraints.
• Resource-free states and resource quantifier: Semi-Gibbs states YS12 = YS1⊗YS2 with Yx = exp(−βxHx)/Zx form the free set TS12. Thermodynamic resource is quantified by α-free entropies Sα(ρS1⊗YS2 | YS1⊗YS2) = Dα(ρS12 || YS1⊗YS2) − log(Z1Z2), with Dα the Rényi α-relative entropy.
• Second laws (Theorem 2): For block-diagonal states ([ρS12, HS12]=0), a CSLTO-induced transformation (ρS12, HS12)→(ρ′S12, HS12′) is possible iff Sα(ρS1⊗YS2 | YS1⊗YS2) ≥ Sα(σS1⊗YS2 | YS1⊗YS2) for all α≥0. These monotones are necessary and sufficient for such transformations and delimit extractable resource/work.
• Work accounting via a battery (Theorem 3): Attach a bipartite ideal battery SW with HS_W = HSW1 + HSW2, initially in its ground state and always in energy eigenstates. The free-entropy distance between initial and final system states equals β1W1 + β2W2, where Wx are the deterministic works stored in the respective battery subsystems. This yields guaranteed one-shot extracted work Wext = W1 + W2 for forward processes and minimal resource cost for reverse (e.g., refrigeration) processes.
• One-step cycle construction: Implement a single-stroke cycle using a SWAP between S1 and S2 and a swap of local Hamiltonians (or equivalently interchanging subsystem-bath couplings between cycles). This compresses isothermal and adiabatic steps into a single CSLTO-consistent transformation.
• Correlation-powered operation: Show that work can be extracted exclusively from inter-system correlations, even when local marginals are thermal, by appropriate non-cyclic or cyclic transformations, quantifying the correlation contribution to free-entropy distance and anomalous heat flows.
• Reversible protocol achieving Carnot efficiency: Provide an explicit two-qubit example with HS1 = α|1⟩⟨1|S1 and HS2 = α|1⟩⟨1|S2, and eigenstate-to-eigenstate swap ρS12 = |0⟩⟨0|S1⊗|1⟩⟨1|S2 → σS12 = |1⟩⟨1|S1⊗|0⟩⟨0|S2 via SLTO. Because α-free entropies coincide for pure eigenstates, the process is reversible in the one-shot sense. Using block-wise conservation of energy and weighted energy, derive the Clausius equality and the efficiency η = 1 − β2/β1 (Carnot).

• Defined semi-local thermal operations (SLTOs) and catalytic SLTOs (CSLTOs) obeying strict conservation of total energy and weighted energy, forming the free operations of a resource theory for quantum heat engines with two baths.
• Established a complete family of second laws (Theorem 2) for block-diagonal transformations under CSLTOs: α-free entropies are monotones, giving necessary and sufficient conditions for feasibility across all α.
• Derived a free-entropy distance identity (Theorem 3): For a CSLTO-driven transformation, β1W1 + β2W2 equals the α-free-entropy difference between initial and final states, yielding guaranteed one-shot deterministic work Wext = W1 + W2 for forward processes and minimal cost for reverse processes.
• Constructed a one-step engine cycle (single stroke) by semi-local interactions that combine isothermal and adiabatic steps, improving one-shot work extraction compared to engines restricted to local thermal operations performed sequentially.
• Demonstrated engines that extract work exclusively from inter-system correlations, quantifying correlation-stored thermodynamic potential in the one-shot regime and explaining anomalous heat flow.
• Presented a reversible one-shot protocol using eigenstate-to-eigenstate swaps in a two-qubit working medium, achieving the Clausius equality β1Q1 + β2Q2 = 0 and Wext = Q1 − Q2 > 0, and attaining the Carnot efficiency η = 1 − β2/β1 despite finite size and one-shot constraints.
• Shown time-translation covariance of SLTOs, implying monotonic decay of coherence between energy eigenspaces and justifying restriction to block-diagonal states after sufficient cycling.
• Argued that CSLTO-based engines generally yield more one-shot extractable work than engines limited to (local) thermal operations, as simultaneous semi-local sub-transformations tighten work bounds.

The results demonstrate that, contrary to prevailing expectations about strong irreversibility and limited deterministic work in the one-shot finite-size regime, carefully designed semi-local interactions with two baths enable reversible one-shot transformations and Carnot efficiency. The resource-theoretic formulation provides a unifying, quantitative language for multi-bath nanoscale engines, connecting work extraction to α-free entropies and clarifying the role of correlations as thermodynamic resources. The one-step cycle compresses traditional four-step Carnot operations into a single stroke, reducing operational overhead and improving one-shot work output relative to sequential local thermal operations. The framework yields both general second-law constraints and constructive protocols, including correlation-powered engines and a reversible eigenstate protocol achieving ηC. Experimentally, SLTOs pose implementation challenges, but related advances in realizing subsets of thermal operations suggest possible routes to approximate or engineered SLTOs. Overall, the work elevates the theoretical limits of quantum engines in regimes where fluctuations are unavoidable, providing pathways for high-efficiency nanoscale thermodynamic devices.

The paper introduces a resource theory for quantum heat engines interacting with two baths in the one-shot finite-size regime, defining SLTOs/CSLTOs and semi-Gibbs states, and deriving complete second-law conditions via α-free entropies. Leveraging these tools, the authors design a one-step engine cycle and explicit reversible protocols that achieve Carnot efficiency, even at finite size and in one-shot settings. They further show that inter-system correlations can be harnessed as a sole fuel for work extraction and that semi-local operations generally enhance one-shot work yield over local thermal operations. Future research directions include: experimental realization or approximation of SLTOs and CSLTOs; extending protocols to multi-level and multi-bath systems with time-dependent Hamiltonians; quantifying and mitigating coherence and control imperfections; exploring autonomous and clock-limited implementations; and investigating robustness under noise, finite bath sizes, and realistic constraints on catalysts and control unitaries.

• Experimental feasibility: SLTOs/CSLTOs require precise global unitaries that conserve both total energy and weighted energy while enabling semi-local interactions; realizing such operations is challenging with current technology.
• State restrictions: Key second-law results are proven for block-diagonal (energy-incoherent) states; coherence is argued to decay under SLTOs, but explicit coherent-state transformations may need additional constraints.
• Idealizations: Assumptions include perfect baths at fixed β1 and β2, exact conservation laws, ideal catalysts that are returned unchanged, and perfect access to SWAP and Hamiltonian updates via clocks.
• Reversibility conditions: Carnot efficiency in one-shot is achieved for eigenstate-to-eigenstate processes; preparing and maintaining such pure eigenstates and precise control may be demanding.
• Scope: While the framework is general, practical performance in noisy, finite-bath, and control-limited settings remains to be quantified; trade-offs between fluctuation suppression and work yield in more complex architectures are not empirically validated here.