Quantum Thermodynamic Processes

• Quantum thermodynamic processes are defined by manipulations of quantum states to control energy, entropy, and information using features like coherence and entanglement.
• Resource theories based on channel capacities and geometric optimization quantify operational attributes such as work extraction, efficiency bounds, and entropy production.
• Finite-time protocols, passivity, and non-commutativity extend classical thermodynamics, leading to innovative designs in heat engines, quantum batteries, and autonomous devices.

Quantum thermodynamic processes comprise the control and transformation of quantum states and operations in ways that manipulate energy, entropy, and information to achieve specific tasks such as work extraction, information erasure, and state interconversion. They extend and refine classical thermodynamics by incorporating uniquely quantum-mechanical features: coherence, entanglement, discreteness, noncommutativity, and memory effects. Recent research has established rigorous operational frameworks—spanning both discrete and continuous regimes—for quantifying fundamental resources and limitations of quantum thermodynamic transformations, notably the roles of coherence and entanglement, passivity and majorization, non-Markovianity, and process capacity. Advances in channel-oriented resource theories and geometric optimization underpin practical protocols for quantum engines, heat baths, autonomous and controlled devices, and information-driven machines. Below, key aspects and methodologies of quantum thermodynamic processes are organized in seven thematic sections.

1. Quantum Speed of Thermodynamic Processes: Coherence and Entanglement

The quantification of the rate at which quantum systems can exchange energy—the "thermodynamic speed"—is central for assessing the operational advantage conferred by quantum resources (Luo, 2024). Luo et al. introduce a rigorous speed metric $S$ for cyclic quantum batteries, built from the instantaneous rate of change of a Hellinger-type energy distance: $$S = \max_H \left. \frac{d}{dt} D_E(t, t') \right|_{t' = t}, \quad D_E(t, t') = \sqrt{ \left( \sqrt{E_{U_t}} - \sqrt{E_{U_{t'}}} \right)^2 },$$ where %%%%1%%%% is the extractable energy under cyclic driving $U_t = e^{-i H t}$.

For pure states and optimal driving, this reduces to the Hamiltonian variance: $S^2 = \langle\psi|(\Delta H)^2|\psi\rangle.$

The central findings can be summarized as follows:

• Quantum coherence: Any pure state with coherence (off-diagonal elements in the energy basis) allows strictly greater thermodynamic speed than any incoherent (diagonal) state. This is formalized by showing $S(\text{coherent}) > S_{\rm ic}$, where $S_{\rm ic}$ is the optimal speed over all incoherent states.
• Genuine multipartite entanglement: States with genuine $n$-body entanglement (e.g., GHZ, Dicke) can exceed the maximal "biseparable" speed $S_{\rm bs}$, with scaling that is quadratic in the number of constituents (e.g., $O(N^2)$), surpassing any product or separable mixture benchmarks.
• Thermodynamic witness: Surpassing the biseparable ceiling $S_{\rm bs}$ serves as an entanglement witness, providing operational evidence of entanglement without requiring state tomography.

This framework unifies the roles of coherence and entanglement as operational resources that fundamentally accelerate energy exchange in quantum thermodynamic cycles.

2. Discrete Quantum Trajectories and Operational Laws

Discrete quantum processes can be formalized as sequences of "thermodynamic configurations"—pairs of quantum states and Hamiltonians—linked by two primitive transformations: discrete unitary transformations (DUTs, representing isolated work steps) and discrete thermalizing transformations (DTTs, representing heat exchange at fixed Hamiltonian) (Anders et al., 2012).

Every trajectory built from DUTs and DTTs obeys a discrete second law: $$\Delta S = S(\rho_N) - S(\rho_0) \geq \Lambda(\mathcal{T}),$$ where

$$\Lambda(\mathcal{T}) = \sum_{k=0}^{N-1} \beta_{k+1} Q_{k \rightarrow k+1}$$

provides a direct analogue of Clausius' inequality.

• Reversible limit: Refined trajectories with an infinite number of ultrafine steps saturate the bound, recovering reversibility.
• Discrete Carnot cycle: Four-stroke cycles built with DUTs/DTTs reproduce Carnot efficiency, operationally linking discrete and continuous thermodynamic protocols.

This formalism enables the consistent definition and calculation of heat and work for arbitrary quantum discrete trajectories, uniting equilibrium and non-equilibrium scenarios.

3. Passivity, Majorization, and Virtual Temperatures

Passivity is a condition on quantum states ensuring that no work can be extracted under cyclic unitary evolution; passive states are diagonal in the energy basis with non-increasing populations (Sonkar et al., 8 Jan 2026). The concept of "virtual temperature" $T_i$ for adjacent energy levels ($i, i+1$) is introduced: $$T_i = \frac{\omega_i}{\ln p_i - \ln p_{i+1}}, \quad \omega_i = E_{i+1} - E_i,$$ allowing characterization of heat-flow direction and the ordering of state transformations via majorization.

• Mean virtual temperature: The spectrum $\{T_i\}$ can be aggregated into a mean $\widetilde{T}$, bounded between the minimum and maximum virtual temperatures ($T_{\min} \leq \widetilde{T} \leq T_{\max}$).
• Majorization and efficiency bounds: The performance of quantum heat engines (e.g., Otto cycles) is upper-bounded by functions of these virtual temperatures. For example, the maximal efficiency for the Otto cycle is

$\eta_{\rm ub} = 1 - \frac{T_{\min}}{T_h},$

which can fall below the Carnot limit.

This framework provides spectrum-only bounds for quantum engines and links resource-theoretic passivity and majorization to thermodynamic performance.

4. Quantum Thermodynamics of General Quantum Processes and Channels

Quantum processes are generally modeled as completely positive, trace-preserving (CPTP) maps ("quantum channels"). Operational thermodynamics can be formulated for arbitrary CPTP maps via channel-based resource theories (Faist et al., 2018, Badhani et al., 14 Oct 2025, Faist et al., 2019, Binder et al., 2014).

• Thermodynamic capacity $C_{\rm th}(\mathcal{N})$: For any channel $\mathcal{N}$, the maximal (single-letter, additive) difference of free energy across the channel defines its thermodynamic capacity: $$C_{\rm th}(\mathcal{N}) = \max_\rho [F(\mathcal{N}(\rho)) - F(\rho)],$$ with $F(\rho) = \Tr[H \rho] - kT S(\rho)$.
• Reversibility: In the macroscopic (i.i.d.) limit, this capacity characterizes both the minimal work cost for simulating one channel from another and the maximal extractable work.
• Channel free energy via relative entropy: Badhani–Das–Das (Badhani et al., 14 Oct 2025) define channel free energy as

$$F^\beta(\mathcal{N}) = \beta^{-1} D[ \mathcal{N} \| \mathcal{T}^\beta ],$$

where $\mathcal{T}^\beta$ is the "absolutely thermal" replacer channel.

• Operational interpretations: Channel free energy quantifies the one-shot and asymptotic rates for erasure (work extraction), purity distillation, and randomness generation.
• Passivity, ergotropy, and operational heat: The change in passive spectrum (majorization) dictates positivity of operational heat and entropy change.

Channel-based frameworks generalize classical laws and provide reversible resources for designing, transforming, and benchmarking quantum devices.

5. Finite-Time Thermodynamics, Geometric Optimization, and Irreversibility

Quantum processes in finite time exhibit irreversibility owing to rapid driving and non-equilibrium effects (Rolandi, 2024, Abiuso et al., 2020). Geometric thermodynamics introduces a Riemannian metric tensor $g_{ij}$ (thermodynamic metric) on the control-parameter manifold, yielding:

• Thermodynamic length and entropy production: $$\Sigma = \int_{0}^\tau \dot{\lambda}^i(t) \, \dot{\lambda}^j(t) \, g_{ij}(\lambda(t)) dt,$$ with minimal dissipation scaling as $\Sigma^* \propto L^2/\tau$, where $L$ is the geodesic length.
• Geometric bounds: The total entropy production obeys refined inequalities: $\Delta S \geq L^2 / \tau,$ representing quantum analogues of classical thermodynamic-length/“horse–carrot” bounds.
• Finite-time Landauer principle: Information erasure in finite time requires excess dissipation beyond the reversible ($k_B T \ln 2$) footprint, with corrections

$Q \geq k_B T \Delta S + a \frac{\hbar}{\tau},$

where $a$ is a model-dependent constant and $\tau$ is the protocol duration, highlighting Planckian speed limits.

• Collective effects: Multi-component systems can share losses, accelerating approach to the reversible regime, but total dissipation remains positive, consistent with the third law.

Geometric methods allow optimal protocol design with performance bounds for mesoscopic engines and information-driven devices.

6. Non-Markovianity as a Thermodynamic Resource

Non-Markovian (memoryful) quantum processes—formally described via process tensors or quantum combs—present further avenues for thermodynamic advantage (Zambon et al., 2024). The amount of non-Markovianity can be quantified via minimal relative entropy distance to the Markovian set: $$N(\Upsilon) = \min_{\Xi\, \text{Markovian}} \max_S S[\Upsilon(S) \| \Xi(S)],$$ and manifests operationally in three mechanisms:

• Work investment: Joint optimization across steps enables spending work early for increased extraction later ($N^{1/4}$-law scaling).
• Multitime correlations: Storing and leveraging time-correlated outputs boosts extraction ($N$-law).
• System–environment correlations: Full optimization (quantum comb) accesses latent correlations, yielding an additional advantage ($N^{1/4}$-law).

Each mechanism lifts the sequential protocol limit and provides continuity-bounded increases in distillable work.

7. Quantum Heat Engines, Cycles, and Phase-Space Modifications

Quantum heat engines—the operationalization of thermodynamic cycles—display distinctive behaviors in the quantum regime, especially when the system's phase space admits non-commutative extensions (Pandit et al., 2019). For coupled oscillators, Otto and Stirling cycles in non-commutative space ($NC_\theta, NC_{\gamma,\xi}$) exhibit:

• Efficiency boosting ("catalytic effect"): Efficiency increases with phase-space non-commutativity, often exceeding classical and commutative-space bounds; the Stirling cycle is particularly enhanced, approaching the ideal limit.
• Analysis by spectrum engineering: Modified energy levels (normal mode splitting, density of states changes) increase extractable work and the density of thermodynamic resources.

Experimental realization of non-commutative effects (via engineered resonators or ion traps) is suggested as a pathway for surpassing standard engine performance.

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Quantum thermodynamic processes thus comprise a unified operational framework linking quantum control, resource theory, information, geometry, and non-equilibrium phenomena. Recent advances have established bridges between coherence, entanglement, passivity, and non-Markovianity as central resources; developed geometric and channel-based quantifiers of thermodynamic value; and identified protocol design principles for deterministic and stochastic device operation. The field continues to refine the microscopic and operational underpinnings of work, heat, and entropy dynamics in the quantum regime.