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The role of quantum information in thermodynamics --- a topical review

John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, Paul Skrzypczyk

arXiv:1505.07835v3quant-ph

TL;DR

This topical review addresses how quantum information theory can clarify the foundations and operation of quantum thermodynamic systems. It synthesizes resource-theoretic, entanglement-based, fluctuation-theorem, and thermal-machine approaches, while highlighting that realistic dynamics, system-specific constraints, and scope boundaries shape the conclusions. The result is an interdisciplinary overview intended as an entry point rather than a comprehensive account.

  • Problem

    Quantum thermodynamics must connect microscopic quantum dynamics with thermodynamic behavior while accounting for quantum correlations, realistic constraints, and non-equilibrium fluctuations.

  • Method

    The paper provides a topical synthesis of quantum-information approaches to statistical mechanics, resource theories, entanglement, fluctuation theorems, and thermal machines.

  • Results

    The review shows how typicality, entanglement, fluctuation relations, and resource-theoretic protocols connect quantum-information concepts with thermodynamic behavior.

  • Takeaways & Limitations

    Quantum information offers a framework for analyzing small quantum systems and interdisciplinary quantum-thermodynamic questions, provided physical and dynamical restrictions are respected.

Abstract

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This topical review article gives an overview of the interplay between quantum information theory and thermodynamics of quantum systems. We focus on several trending topics including the foundations of statistical mechanics, resource theories, entanglement in thermodynamic settings, fluctuation theorems and thermal machines. This is not a comprehensive review of the diverse field of quantum thermodynamics; rather, it is a convenient entry point for the thermo-curious information theorist. Furthermore this review should facilitate the unification and understanding of different interdisciplinary approaches emerging in research groups around the world.

I. INTRODUCTION

This topical review examines how quantum information theory and thermodynamics illuminate one another, focusing on foundations, resource theories, entanglement, fluctuation theorems, and thermal machines. It is designed as an entry point for information theorists while surveying interdisciplinary approaches rather than the entire field.

  • Quantum information: Quantum information introduces distinctive features—including limited readout, no-cloning, and nonlocal correlations—that require extensions beyond classical information theory.These features motivated the development of quantum information as a distinct discipline.
  • Resource theories: Quantum information extends thermodynamic analysis to small quantum systems and revisits thermodynamic foundations.Resource-theoretic approaches organize constraints such as energy conservation and locality, and can combine them by restricting allowed operations.
  • Foundations: Information-theoretic methods contribute to statistical mechanics through principles such as Jaynes’s maximum-entropy approach.The review connects these ideas to deriving thermodynamics from microscopic classical or quantum mechanics.
  • Non-equilibrium thermodynamics: Non-equilibrium research links equilibrium thermodynamic quantities to non-equilibrium ones through fluctuation theorems and quantum-information techniques.Examples include phase-estimation methods for work and heat statistics and feedback fluctuation theorems for Maxwell’s demons.
  • Thermal machines: The review discusses intrinsic quantumness in machines operating at and below the quantum threshold using quantum-information tools.It reviews recent progress on the advantages and disadvantages of such machines.
  • Scope and aims: The review studies the interplay between quantum information and thermodynamics across several current research directions.Its topics include statistical-mechanical foundations, resource theories, thermodynamic entanglement, fluctuation theorems, and thermal machines.
  • Scope and related reviews: The topical scope excludes areas already covered extensively elsewhere, including broad treatments of equilibration, thermal machines, Maxwell’s demon, and quantum thermodynamics.The authors point readers to other reviews and books for these topics.

II. FOUNDATIONS OF STATISTICAL MECHANICS

The review explains how quantum-information typicality arguments replace the equal-a-priori-probabilities postulate for accessible subsystems, while emphasizing that physically realizable dynamics and system-specific Hamiltonians constrain this picture. It connects equilibration and thermalization to entanglement, concentration of measure, and circuit-based notions of typicality.

  • Foundations: Entanglement reconciles constant global von Neumann entropy under unitary evolution with increasing entropy in observed subsystems.Subsystem entropy grows because the subsystem becomes entangled with the rest of the universe.
  • Typicality: Typicality replaces equal a priori probabilities by showing that most global states yield nearly indistinguishable reduced states for accessible subsystems.For Haar-random states in a restricted subspace, distinguishability from the reference state decreases exponentially with the subspace dimension.
  • Typicality: The trace norm quantifies distinguishability between reduced states through the largest observable expectation-value difference under a bounded operator norm.The typicality estimate relies on concentration of measure and Levy’s lemma.
  • Physical restrictions: Uniform typicality over an entire Hilbert subspace is physically questionable because local Hamiltonian evolution cannot generate most states within polynomial time.This motivates typicality analyses over more realistic state sets, including matrix product states and physically generated states.
  • Circuit typicality: Random local quantum circuits provide physically relevant approximate designs, and almost-2-designs can decouple a subsystem from its environment independently of the environment’s size.These results connect circuit dynamics and decoupling to generalized typicality.
  • Dynamical typicality: Integrable systems can violate predictions based on generic-state typicality because their trajectories do not spend most times near the completely mixed state.Equilibration and thermalization therefore depend on dynamical properties, including the Hamiltonian and initial state.

B. Equilibration. Maximum entropy principle from quantum dynamics

The review explains how quantum dynamics can produce equilibration and thermalization, connecting equilibrium states to entropy maximization while identifying conditions and boundaries for these results.

  • Equilibration: Equilibration asks how reversible unitary dynamics drive a system toward a persistent state.
  • Equilibration: For subsystems, equilibration is defined by a small time-averaged trace distance from the time-averaged state, making the subsystem indistinguishable from equilibrium for most times.
  • Equilibration: Non-degenerate energy gaps, or the weaker condition of no hugely degenerate gaps, provide sufficient conditions for equilibration when the initial state spans many energies.
  • Maximum entropy principle: The equilibrium state maximizes von Neumann entropy subject to the conserved quantities, making the maximum entropy principle a consequence of quantum dynamics.
  • Thermalization: Thermalization requires explaining why equilibrium is described by a Gibbs state largely independent of initial details, while integrable systems instead require a generalized Gibbs ensemble.
  • Thermalization: The framework is limited by interaction strength and system structure: weak coupling can fail for large subsystems, and diagonal-ensemble descriptions may require exponentially many conserved quantities.

D. Equilibration times

Equilibration is known to occur in many settings, but its timescale remains poorly controlled and depends on the Hamiltonian, observables, and system properties.

  • Open problems: The main open challenge is determining equilibration timescales, which may be extremely long or absent in systems such as glasses.
  • Known bounds: General equilibration-time bounds are rigorous but scale exponentially with system size, whereas short times have been proven only for selected observables, Hamiltonians, and initial states.
  • Open problems: Further work must identify which Hamiltonian and observable features produce physically reasonable equilibration times.
  • Open problems: Typicality should be extended to symmetric states, since physical Hamiltonians are not generally generic.
  • Open problems: A satisfactory quantum notion of integrability is still lacking, hindering classification of many-body systems with distinct dynamical behavior.
  • Relative thermalization: Local Gibbs form alone does not ensure that a subsystem can serve as a thermal bath; relative thermalization also requires absence of correlations with the reference system.

III. RESOURCE THEORIES

Resource theories describe thermodynamics operationally by treating thermal states as free resources and organizing which state transformations are allowed under specified constraints.

  • Thermodynamics as a resource theory: Thermodynamic resource theories treat equilibrium states as free resources and catalysts as reusable aids for state transformations.
  • Resource-theory framework: A resource theory fixes a state space and allowed operations, which induce a preorder describing possible transformations between states.
  • Resource-theory framework: Necessary and sufficient transformation conditions can be expressed through functions, while monotones provide necessary conditions that cannot increase under allowed transformations.
  • Thermodynamic models: Thermodynamic models commonly allow contact with thermal baths and reversible operations preserving thermodynamic quantities, with variants determined by the operation set.
  • Noisy operations: In the fully degenerate case, noisy operations combine adjoining maximally mixed systems, unitaries, and subsystem discarding.
  • Noisy operations: For this setting, noisy operations and unital maps induce the same preorder, and majorization is necessary and sufficient for state transformations.
  • Noisy operations: Von Neumann entropy is a Schur monotone: if ρ majorizes σ, then S(ρ) is smaller than S(σ).

2. Thermal operations

Thermal operations model quantum thermodynamics using Gibbs-state ancillas and energy-conserving transformations, with free energies and related quantities governing state conversion.

  • Thermal operations: For non-degenerate Hamiltonians, equilibrium states are Gibbs states, and thermal contact is modeled by adjoining Gibbs-state systems.
  • Thermal operations: Thermal operations use energy-conserving unitaries, while free-energy variants provide monotones for state transformations.
  • Gibbs-preserving maps: Gibbs-preserving maps agree with thermal operations on classical state preorders but are not operationally defined for general achievability claims.
  • Gibbs-preserving maps: Thermal operations cannot create coherence, whereas Gibbs-preserving maps can enable transformations that thermal operations cannot achieve.
  • Coherence: Coherence reservoirs can catalytically support coherent operations, although their state spreads with use and realistic reservoirs eventually require recharging.
  • Thermal operations: Free energy is recovered as a monotone in the single-shot regime, alongside a family of monotones based on quantum Rényi relative entropies.
  • Coherence: In the many-copy limit, work yields with and without a coherence reservoir converge, so such reservoirs are mainly critical for single-shot small systems.

5. Catalysts

Catalysts can enable otherwise forbidden state transformations, but unrestricted catalysts trivialize the constraints; energy and dimension bounds restore meaningful thermodynamic monotones. The section also highlights unresolved costs of coherent catalysts, realistic clocks, and free Gibbs states.

  • Catalytic transformations: For any finite ε and states ρ and σ, an unrestricted, sufficiently large catalyst can enable ρ → σ even when thermal operations forbid it.These catalysts are thermodynamic analogues of embezzling states in LOCC.
  • Catalytic transformations: Energy and dimension restrictions on catalysts recover familiar monotones for state transformations, including free-energy constraints when ε ∝ n^-1.The restriction links catalyst size and approximation error to thermodynamic state-conversion conditions.
  • Clocks and implementation costs: Thermal resource theories treat energy-preserving unitaries as free, although implementing them requires time-dependent control, precise clocks, and accounting for clock work costs.Explicit-clock protocols use a clock and a weight as energy and coherence reservoirs.
  • Clocks and implementation costs: The explicit-clock construction remains idealized because its clock is infinite-dimensional and modeled as a relativistic particle.Whether realistic systems can realize these clock properties remains open.
  • Free states and passivity: Gibbs states are completely passive: no work can be extracted from any number of copies using energy-conserving unitaries, making them suitable free states.This supports allowing arbitrary thermal subsystems for free, provided another nonequilibrium resource is present for work extraction.

9. Finite-size effects

Finite-size and single-shot settings refine thermodynamic quantities away from the idealized large-bath, many-copy regime. Resource-theoretic formulations connect work, correlations, and allowed transformations while exposing broader questions about realistic operations.

  • Finite baths: Restricted heat baths alter the precision of state transformations and yield effective work-cost measures that converge to standard quantities as bath size grows.Examples include limits on energy gaps or on the number of qubits that can be thermalized.
  • Finite-size regimes: Many-copy studies recover familiar monotones such as von Neumann entropy and free energy, whereas single-shot work addresses individual quantum systems.The many-copy regime approximates large uncorrelated systems.
  • Single-shot work: Single-shot analyses treat work as a random variable, extending average-work descriptions to fluctuations around thermodynamic transformations.Resource theories represent work storage with an explicit system acted on jointly with the system of interest.
  • Definitions of work: A harmonic oscillator can serve as a quantum work-storage system, whose final reduced state reveals average work, fluctuations, coherences, and system–storage correlations.Other resource theories can use alternative reservoirs, such as spin-based angular-momentum stores.
  • Landauer’s principle: Landauer erasure costs kBT ln 2 for a completely unknown bit in the ideal infinite-bath, many-operation limit, with finite-size effects requiring separate analysis.For correlated quantum memory, smooth conditional max entropy can become negative, allowing work gain at the cost of correlations.

4. Resource theories of knowledge

The review connects quantum-information resource theories with thermodynamic and locality constraints, using them to analyze entanglement, correlations, and knowledge across quantum systems. Entropic restrictions bound which correlations can be created and motivate resource theories that combine multiple constraints.

  • Resource theories of knowledge: Quantum-information methods provide a resource-theoretic account of thermodynamics from thermal-state emergence to state manipulation under energy-conserving unitaries.The approach targets limitations on transformations imposed by the investigated system’s physical constraints.
  • Hybrid resource theories: Combining thermodynamic and locality constraints changes which quantum-information resources can be produced and how thermodynamic quantities behave at the quantum scale.The review identifies adaptive resource theories as a route toward quantifying resources under multiple restrictions.
  • Entropic restrictions: States sufficiently close to the maximally mixed state cannot be entangled by unitary transformations because their distance from that state is invariant.This provides a temperature- and entropy-related obstruction relevant to finite-temperature quantum computation.
  • Separability from spectrum: For 2×m-dimensional states, the spectrum-based condition for possible entanglement is both sufficient and necessary.The criterion concerns whether entanglement exists somewhere in the state’s unitary orbit.
  • Correlations and entanglement: Mutual information quantifies correlations in unitary orbits, where global unitaries can achieve Iρ(A : B) = 2 log2(d) − S(ρ).This offers an operationally clear alternative when entanglement measures lack a unique general currency.
  • Correlations and entanglement: Genuine multipartite entanglement can be created from multipartite thermal states when kBT/E < n/(2 ln(n)) + O(n/ln(n)^2).The cited protocol relates the entanglement threshold to subsystem dimension and an energy scale.

B. Correlations and entanglement in a thermodynamic background

Thermodynamic transformations reveal that correlations and entanglement can carry work value, but the advantage depends on energy costs, locality restrictions, system size, and the chosen work measure.

  • Thermodynamic value of correlations: Accounting for average energy changes reveals an intrinsic work value for correlations and entanglement.This extends analyses that assume unlimited external energy or fully degenerate Hamiltonians.
  • Thermodynamic value of correlations: Thermal operations can reduce target entropy using a heat bath, but ignoring energy costs makes quantum resource production trivial.Free-energy accounting instead assigns intrinsic work value to correlations and bounds the free-energy cost of entanglement.
  • Quantum versus classical correlations: For two qubits, entangled correlations can store twice as much extractable work as the best separable or classical correlations.The comparison concerns work stored purely in correlations, while the quantum advantage vanishes in the thermodynamic limit.
  • Creating entanglement: Creating entanglement from thermal states requires an energy investment, illustrated by a two-qubit concurrence expressed in terms of invested average energy.The simplest example considers two qubits with energy gap E at zero temperature.
  • Feedback and locality: Quantum feedback control studies work costs from measurement-induced correlations between a system and its memory.The net-work bound includes mutual-information terms and the free-energy change.
  • Feedback and locality: Locality restrictions reduce extractable work, with the resulting work deficit connected to entanglement measures for pure states.For pure states, the relevant bound coincides with entanglement of formation or equivalent measures reducing to marginal entropy.

D. Entanglement resources in thermodynamic tasks

Entangling operations can enable work extraction from passive states, while entanglement itself need not be generated; thermodynamic constraints and locality restrictions jointly shape quantum information processing.

  • Work extraction from passive states: Global entangling unitaries enable work extraction from copies of passive states when local unitaries cannot extract work.Here, the entangling power of the unitary is treated as a resource for work extraction.
  • Work extraction from passive states: The required entangling operation does not imply that the extraction process generates entanglement.The procedure can be implemented dynamically without generating entanglement, although the most direct transformation may substantially entangle the systems.
  • Work extraction from passive states: Cyclic operations can provide a quantum advantage in charging power when work per unit time is the figure of merit.This concerns power rather than only the total extractable work.
  • Thermodynamic signatures of entanglement: Thermodynamic observables may reveal underlying entanglement, directly connecting quantum thermodynamics with entanglement theory.The review highlights this possibility in the context of quantum many-body physics and thermodynamic observables.
  • Scope and open directions: Resource theories capture limitations imposed by particular physical constraints, so combining thermodynamic and locality restrictions remains an open direction.A consistent adaptive resource theory could clarify the role of genuine quantum effects such as entanglement.
  • Quantum fluctuation relations: Stochastic thermodynamics treats work and heat in small systems as random variables characterized by probability distributions.This framework supports fluctuation theorems beyond the linear-response regime.
  • Quantum fluctuation relations: The quantum work distribution is built from energy measurements before and after a driven unitary protocol, capturing thermal and quantum measurement fluctuations.The protocol begins in a Gibbs state and generally ends in a non-equilibrium state.
  • Quantum fluctuation relations: The Tasaki-Crooks relation connects forward and backward work fluctuations in arbitrary non-equilibrium transformations to an equilibrium free-energy difference.The backward process uses the time-reversed protocol from the final Gibbs state.

B. Phase estimation schemes for extraction of quantum work and heat statistics

Quantum-information protocols provide ways to obtain work statistics beyond direct projective measurements, and laboratory tests have demonstrated characteristic-function extraction in quantum systems.

  • Motivation: Quantum work statistics were experimentally inaccessible until quantum-information methods enabled their acquisition for quantum systems.The review identifies this as a major contribution of quantum information to statistical mechanics.
  • Phase estimation: Phase-estimation schemes use an ancilla and tomography to obtain the characteristic function, from which work statistics are extracted by Fourier transformation.The approach is related in spirit to the DQC1 algorithm.
  • Experimental demonstration: Liquid-state NMR experiments first tested characteristic-function measurements and demonstrated work fluctuation theorems and quantum work statistics.The proposals also motivate extensions to open-system dynamics beyond weak coupling.
  • Alternative measurement schemes: POVM-based schemes can sample the work distribution with a single measurement in time using an appropriate ancilla.The method was proposed for estimating free energy on a quantum computer and later developed toward implementation with ultracold atoms.

C. Fluctuation relations with feedback, measurement and CPTP maps

Feedback and measurement generalize quantum fluctuation relations by incorporating information acquired during control, while relative entropy links irreversible entropy production to quantum correlations.

  • Feedback control: Feedback control extends fluctuation-theorem frameworks by conditioning thermodynamic operations on measurement outcomes.The quantum setting includes an initial energy measurement, outcome-dependent Hamiltonian changes, and a final energy measurement.
  • Feedback control: The feedback relation introduces a measurement-dependent term alongside work and free-energy change.Its value quantifies how effectively the demon uses acquired information, with values below or above one indicating inefficient or efficient feedback.
  • Feedback control: Quantum feedback fluctuation theorems incorporate mutual information between recorded outcomes and true measurement results, and can include an explicit memory system.These extensions also illuminate thermodynamic work obtainable from entanglement.
  • Entropy production and correlations: Relative entropy is central to non-equilibrium quantum thermodynamics because of its close relationship with quantum-state free energy.It also serves as a central quantity in quantum information descriptions of entanglement and correlations.
  • Entropy production and correlations: Dissipated work and irreversible entropy production have been connected to entanglement witnesses and to correlations generated in quantum systems.These relationships include oscillator chains and open-system settings.
  • Outlook: Quantum fluctuation theorems are exact for arbitrary non-equilibrium dynamics and are being applied to energy, heat, and information transport in quantum technologies.Applications beyond quantum thermodynamics remain an emerging research direction.
  • Outlook: Quantum phase estimation has already extracted work statistics from a small non-equilibrium quantum system, motivating extensions to non-passive, many-body, and open-system scenarios.These extensions are presented as future avenues rather than established results.
  • Outlook: Initial steps toward unifying work statistics, fluctuation theorems, and single-shot statistical mechanics have been reported.The review anticipates further links among these approaches.

VI. QUANTUM THERMAL MACHINES

Quantum thermal machines extend heat engines and refrigerators to small quantum systems, where entanglement, discord, coherence, and correlations can affect operation and generate resources.

  • Overview: Quantum machines can be built from a single qutrit or from two or three qubits.The review discusses machines operating with thermal reservoirs and considers their quantum-information properties.
  • Stationary behaviour: Absorption refrigerators use cold, hot, and work reservoirs, with the hot reservoir supplying energy without external work.The three-qubit model uses resonant energy levels and an interaction that mediates energy transfer.
  • Stationary behaviour: The coefficient of performance is COP = QC/QH, and unlike efficiency it can exceed 1.COP compares heat extracted from the cold reservoir with heat supplied by the hot reservoir.
  • Stationary behaviour: Near the maximal Carnot limit, the stationary state is fully separable, whereas farther from that regime all forms of multipartite entanglement can occur.The entanglement is small, as expected under weak inter-qubit coupling.
  • Stationary behaviour: Entangled refrigerators can outperform separable ones, with the cooling advantage depending only on entanglement across the R|CH partition.This partition separates energy entering the machine from energy leaving it, linking entanglement to transport properties.

2. Transient behaviour

Transient dynamics reveal additional cooling and quantum advantages beyond stationary operation. Coherent oscillations, initial coherence, and engineered reservoirs can lower temperatures or improve machine performance.

  • Transient behaviour: Strong inter-qubit coupling produces Rabi oscillations with period approximately 2π/g, enabling transient cooling below the stationary temperature.Weak coupling is effectively overdamped, while non-Markovian memory produces more complicated transients that can also reach colder temperatures.
  • Transient behaviour: Appropriate couplings can keep the system below its stationary temperature for a long time without oscillating above it.The stable preparation regime requires, in particular, the weakest coupling to be the hot reservoir.
  • Transient behaviour: Small initial coherence can induce temperature oscillations even under weak-interaction dynamics, allowing cooling below the stationary temperature.In stronger-interaction and non-Markovian cases, initial coherence also increases oscillation magnitude.
  • Transient behaviour: Transient operation can generate considerably more entanglement than the stationary regime across R|CH and in genuine multipartite form.R|CH corresponds to the partition between energy entering and energy leaving the machine.
  • Reservoir engineering: Squeezed reservoirs modify the hot-bath dynamics so that refrigerators can exceed the ordinary Carnot COP bound.The comparison uses the standard baths at βC, βR, and βH without reservoir engineering.
  • Reservoir engineering: A squeezed reservoir changes a quantum Otto engine’s thermalization stage by replacing the standard hot reservoir.The engine uses a time-dependent cycle with expansion and thermalization stages.
  • Reservoir engineering: The squeezed-reservoir results do not violate the second law because they lie outside the ordinary Carnot regime of applicability.The effective hot-reservoir temperature is modified by squeezing.

C. Quantum thermodynamic signatures

Quantum thermodynamic signatures distinguish genuinely quantum operation from effects attributable only to discrete energy levels. The review highlights power witnesses, entanglement generation, and coherence as promising signatures and identifies open experimental and coupling-regime questions.

  • Quantum thermodynamic signatures: A classical comparison machine is constrained to population dynamics, unchanged energy levels and couplings, and no additional heat or work sources.Pure dephasing in the energy eigenbasis provides one way to impose these constraints.
  • Quantum thermodynamic signatures: For small engine action, a state-independent classical power bound scales with the cycle duration τcyc.Engine action is the product of duration and an energy scale defined through operator norms in the master equation.
  • Quantum thermodynamic signatures: Quantum engines can provably outperform corresponding classical machines, with power an order of magnitude larger in the demonstrated regime.The comparison removes additional dephasing from the quantum engine while preserving the model’s energy structure and couplings.
  • Quantum thermodynamic signatures: Entangled steady states in autonomous machines provide a route to generating entanglement through dissipative interactions with thermal environments.A machine may generate steady-state entanglement even when it functions only as a bridge carrying heat between reservoirs.
  • Quantum thermodynamic signatures: Two-qubit bridge implementations suggest that experimentally accessible machines can maintain steady-state entanglement at potentially distillable levels.This proposal focuses on entanglement generation rather than a conventional thermodynamic task.
  • Outlook: The section’s main studies focus on weak coupling, leaving the effects of strong bath–machine coupling and the trade-off between noise and driving unresolved.Stronger coupling may damage fragile quantum correlations, while stronger driving may amplify quantum effects.
  • Outlook: Further work should develop quantum signatures that apply across broader scenarios and demonstrate them experimentally.The review also suggests examining how thermal-machine cooling could complement quantum error correction.
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