Thermal Operations in Quantum Thermodynamics

Updated 25 July 2025

Thermal operations are quantum channels defined by coupling a system to a Gibbs-state heat bath and using energy-conserving unitaries to enforce thermodynamic laws.
They establish a resource-theoretic framework where state transitions are governed by thermo-majorization and quantum Rényi divergences, linking classical and quantum aspects.
Applications include analyzing work extraction, engine efficiency, and catalytic protocols, while highlighting the inherent limit in generating or preserving quantum coherence.

Thermal operations comprise a mathematically rigorous and physically motivated class of quantum channels that encapsulate the possible transformations of a quantum system interacting with a thermal environment under fundamental thermodynamic constraints. These operations are defined by coupling the system to a heat bath prepared in a Gibbs state at a fixed temperature and applying an energy-conserving global unitary evolution, after which the bath is traced out. The framework of thermal operations serves as the foundational model in quantum thermodynamics for analyzing work extraction, state transitions, entropy production, coherence dynamics, and engine efficiencies at the nanoscale. Their structure is intimately linked to resource theories of athermality and quantum information, enabling a detailed characterization of what state transformations are physically realizable under strict conservation laws.

1. Definition, Structure, and Physical Principles

A thermal operation (TO) is any quantum channel that can be realized by the following procedure:

Begin with a system $S$ (Hamiltonian $H_S$ ) and an environment $B$ (bath Hamiltonian $H_B$ ) initialized in the Gibbs state $\gamma_B = e^{-\beta H_B} / Z_B$ at inverse temperature $\beta$ .
Apply a joint unitary $U$ that strictly conserves total energy:

$[U, H_S + H_B] = 0$

Discard the bath by tracing over $B$ .

Mathematically, the action of a thermal operation on a state $\rho_S$ is:

$\mathcal{E}_{\mathrm{TO}}(\rho_S) = \operatorname{Tr}_B \left[ U (\rho_S \otimes \gamma_B) U^\dagger \right]$

This construction ensures that (i) the Gibbs state is a fixed point, i.e., $\mathcal{E}_{\mathrm{TO}}(\gamma_S) = \gamma_S$ , and (ii) no external resource other than the thermal bath is required. The structure admits only those operations compatible with both energy conservation and equilibrium preservation, thus anchoring them in the operational constraints of statistical physics (Faist et al., 2014, Ng et al., 2018).

2. Resource-Theoretic Framework and State Transition Criteria

Thermal operations underpin a resource theory of athermality in which free states are Gibbs states and free operations are those that can be realized as TOs. The central question becomes: for given initial and final states $\rho$ and $\sigma$ , is the transition $\rho \to \sigma$ feasible under TOs?

For states diagonal in the energy basis (“incoherent states”), the answer is governed by thermo-majorization (Ng et al., 2018, Czartowski et al., 2023), a partial ordering that generalizes classical majorization:

Each probability vector $p$ (diagonal of $\rho$ ) is reordered according to the ratios $p_i/\gamma_i$ in non-increasing order (the $\beta$ -ordering).
The cumulative curve $f_p^\beta(x)$ is constructed: $(0,0)$ and points $(\sum_{i=1}^k \gamma_{i}, \sum_{i=1}^k p_{i})$ .
A necessary and sufficient condition is that $f_p^\beta(x) \geq f_q^\beta(x)$ for all $x \in [0,1]$ ( $p$ thermo-majorizes $q$ ), ensuring the existence of a Gibbs-preserving stochastic map connecting $p$ to $q$ .

For general (possibly coherent) quantum states, the family of generalized second laws—expressed via quantum Rényi divergences $D_\alpha(\rho\|\gamma)$ —must not increase under TOs. In the i.i.d. (many-copy) limit, all Rényi divergences coalesce into the standard quantum relative entropy, and TOs recover the classical free energy law (Ng et al., 2018, Sagawa et al., 2019).

3. Quantum Coherence and the Gap to Gibbs-Preserving Maps

A salient feature distinguishing quantum from classical thermodynamics in this framework is the handling of quantum coherence between energy eigenstates. Thermal operations, by virtue of their energy-conserving construction, cannot generate or transfer coherence between different energy subspaces: if the input state is diagonal, so is the output (Faist et al., 2014, Hu, 2018). This stems from the covariance of TOs under time translations generated by $H_S$ . By contrast, the more general class of Gibbs-preserving maps (GPMs), which simply require $\mathcal{E}(\gamma_S) = \gamma_S$ , can, in principle, create and manipulate coherence, as they are not constrained by energy conservation (Faist et al., 2014).

This strict limitation means that the set of state transitions possible under TO is strictly smaller than that achievable under GPMs when coherence is relevant. For example, the transformation $|1\rangle \to |+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$ (a maximally coherent superposition) is forbidden under TOs if starting from an energy eigenstate, but is allowed under GPMs. These distinctions have profound implications for the structure of the resource theory and the formulation of second laws that govern coherent state transformations.

4. Variants, Universality, and Experimental Realization

Numerous operational subsets and extensions of TOs have been proposed to bridge theory with laboratory feasibility and to analyze the role of memory:

Elementary Thermal Operations (ETOs): Restrict TOs to act non-trivially on at most two energy levels at a time via sequences of “elementary” d-swaps and T-transforms. Strong (randomized) ETOs are universal only for “quasi-uniform” Gibbs distributions (e.g., in two-level systems or where only one level’s thermal weight differs), while deterministic protocols are generally not universal beyond dimension two (Hack et al., 2023). Algorithms exist to decompose reachable state transitions into sequences of such operations, and the connection to random walks on complete graphs offers further tools for analysis.
Markovian and Coarse Operations: For states diagonal in the energy basis, experimentally feasible operations—namely coarse operations involving only partial thermalizations, level transformations, and at most a single ancillary qubit—are sufficient to implement all possible TOs (Perry et al., 2015, Aguilar et al., 2020, Czartowski et al., 2023). However, without such minimal memory resources, certain state transitions cannot be replicated, establishing that TOs are not generally memoryless. In particular, memory plays a critical role for transitions that “cross” the Gibbs state.
Non-Markovian and Enhanced Thermal Operations: If memory effects are allowed (e.g., through extremal thermal operations or non-Markovian dynamics), both the average performance and stability of quantum heat engines can be improved relative to purely Markovian realizations (Ptaszyński, 2022). However, “enhanced” maps that satisfy only symmetry and Gibbs-preservation (but not necessarily full microscopic implementability) are strictly larger than TO, especially in their coherent sector (Ding et al., 2021).
Collision Models and Finite Complexity: Realistic collision models based on sequential energy-preserving unitaries—each acting with small bath fragments—admit a practical framework for finite-complexity TOs. Even under these constraints, protocols exist for cooling below the bath temperature, with and without small quantum “machines” (ancillas), and quantitative bounds can be placed on achievable cooling and efficiency (Hu et al., 15 May 2025).

5. Application to Work Extraction, Engines, and Thermodynamic Laws

TOs provide operational tools for the quantitative analysis of nanoscale engines and thermodynamic processes:

Work Extraction and Formation: Under TO, protocols involving sequences of energy level manipulations and selective thermalizations (partial or full) can distill work with optimal efficiency as dictated by single-shot and average second law bounds (Perry et al., 2015). The optimal work $W$ is connected to generalized free energies, which are monotonic under TO, and governed by the decrease of suitable Rényi relative entropies between the system and Gibbs states.
Engines and Resource Advantage: Comprehensive frameworks for resource-based engines formalize engine efficiency in terms of monotones (such as the “free advantage”) and identify analytic bounds using tree-states, achievable by protocols of two-level TOs (Czartowski et al., 8 Jul 2025).
Catalysis and Hierarchy Collapse: The inclusion of catalysts—auxiliary systems returned unchanged—collapses the operational gap between full TOs and various experimentally accessible subsets, such as elementary or Markovian operations, even when initial states contain quantum coherence (Son et al., 2023).

6. Mathematical Structure, Continuity, and Hierarchies

The mathematical formulation of TOs is closely tied to the structure of the system and bath Hamiltonians:

Resonant Spectrum and Bath Engineering: TOs can be realized—up to closure—using only bath Hamiltonians that have spectrum resonant with the system, i.e., their energy differences match the Bohr spectrum of $H_S$ (Ende, 2022). This ensures all possible transitions are accessible and that, especially in qubit systems, the set of TOs can be visualized in low-dimensional parameter spaces, where convexity and other algebraic properties become transparent.
Closure and Convexity: In finite dimensions, the closure of the set of TOs equals the set of enhanced TOs (covariant and Gibbs-preserving maps) for two-level systems. The set admits a faithful semigroup representation, and composition is commutative (Ende, 2022).
Information-Theoretic Characterization: Channels are thermal operations if and only if they admit a unitary dilation that leaves the environment invariant when applied to the equilibrium state, encapsulating the "informational zeroth law" of thermodynamics (Lie et al., 22 Jul 2025). This criterion uniquely distinguishes TOs within the hierarchy of Gibbs-preserving and catalytic channels.

7. Implications and Limitations

Thermal operations occupy a foundational role in quantum thermodynamics and set the operational boundary for physically implementable, resource-free quantum channels respecting thermodynamic laws:

Limitations: They cannot, in general, generate or manipulate coherence between energy levels, nor can they achieve the full set of Gibbs-preserving transformations without additional resources such as time-reference frames or clocks (Faist et al., 2014). Algorithmic cooling is fundamentally limited for continuous-variable (Gaussian) systems under Gaussian TOs, with entropic bounds dictated solely by the colder of the initial or bath temperatures (Serafini et al., 2019).
Irreversibility and Correlations: Entropy production under TO admits an information-theoretic interpretation as the mutual information generated between system and bath in the evolution. For initial states diagonal in energy, no quantum correlations are generated under TO (Dolatkhah et al., 2018).
Avenues for Extension: Addressing TO's limitations for coherent manipulation motivates extensions involving resource-theoretic catalysis, enhanced operations, and careful inclusion of reference frames. Experimental realizations increasingly leverage elementary operations and minimal memory protocols, enabled by insights from both theory and feasible device architectures.

In summary, thermal operations define the maximal set of physically implementable thermodynamic evolutions consistent with energy conservation and equilibrium preservation. Their precise characterization, applicability to a range of thermodynamic tasks, and formal relationship to resource theories, catalysis, and coherence constitute a central pillar in the paper and engineering of quantum thermodynamic processes.