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An introductory review of the resource theory approach to thermodynamics

Matteo Lostaglio

arXiv:1807.11549v2quant-ph

TL;DR

Quantum thermodynamics lacks a self-contained account of how constrained processes order non-equilibrium states and quantum resources. This paper introduces the resource-theory framework and its technical machinery, showing how thermo-majorisation and symmetry constraints organize energetic and coherent transformations. It develops applications including thermal state transformations, coherence-transfer irreversibility, and deterministic work tasks while clarifying the framework’s scope and relation to complementary approaches.

  • Problem

    The paper addresses how to characterize thermodynamic accessibility and refined ordering for non-equilibrium states when equilibrium’s total-ordering assumption does not apply.

  • Method

    The paper gives a self-contained resource-theory introduction, using allowed-process frameworks, thermo-majorisation, and symmetry analysis to study energetic and coherent transformations.

  • Results

    The framework characterizes thermal population transformations through thermo-majorisation and identifies time-translation symmetry as imposing coherence constraints beyond free-energy measures.

  • Takeaways & Limitations

    Resource theory provides a quantum-information language for analyzing non-equilibrium structure, thermodynamic tasks, quantum advantages, and connections with complementary approaches.

  • Takeaways & Limitations

    The framework does not easily handle time-dependent Hamiltonians or quantum effects of macroscopic systems, and its scope within quantum thermodynamics is not generally agreed.

Abstract

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I give a self-contained introduction to the resource theory approach to quantum thermodynamics. I will introduce in an elementary manner the technical machinery necessary to unpack and prove the core statements of the theory. The topics covered include the so-called `many second laws of thermodynamics', thermo-majorisation and symmetry constraints on the evolution of quantum coherence. Among the elementary applications, I explicitly work out the bounds on deterministic work extraction and formation, discuss the complete solution of the theory for a single qubit and present the irreversibility of coherence transfers. The aim is to facilitate the task of those researchers interested in engaging and contributing to this topic, presenting scope and motivation of its core assumptions and discussing the relation between the resource theory and complementary approaches.

INTRODUCTION

The resource theory approach characterizes thermodynamic accessibility under explicitly chosen physical constraints, extending ordering questions from equilibrium to non-equilibrium states. It provides quantum-information tools for analyzing resources, optimal thermodynamic tasks, coherence, and relations among complementary approaches.

  • INTRODUCTION: Thermodynamic accessibility forms a partial order on states modulo reversible interconversion, while entropy helps determine ordering among equilibrium states.The ordering is reflexive, antisymmetric, and transitive when processes can be composed.
  • INTRODUCTION: Non-equilibrium states need not be totally ordered, so no unique extension of entropy or free energy should generally be expected.The total ordering of equilibrium states is the Comparison Hypothesis; it does not generally carry over to non-equilibrium thermodynamics.
  • INTRODUCTION: Resource theories define allowed thermodynamic processes and characterize the partial ordering they induce on non-equilibrium states.This requires refined constraints beyond the standard statement that entropy increases.
  • Scope of the resource theory: The resource theory is not an all-purpose framework: it handles time-dependent Hamiltonians poorly and does not incorporate some macroscopic quantum effects.Its central ordering tools may nevertheless be useful within complementary frameworks.
  • Scope of the resource theory: The framework is most useful for small quantum systems interacting with large or otherwise intractable environments, rather than for exactly solvable dynamics.It can describe certain thermodynamic properties without relying on a master-equation description.
  • Scope of the resource theory: The approach uses quantum-information tools to quantify non-equilibrium properties, study how quantum resources may be harnessed, and identify optimal strategies for thermodynamic tasks.It also provides a language for studying intersections between thermodynamics, information theory, and their quantum counterparts.

Why only thermal states can be free states?

Thermal states are selected as free because they are passive at the background temperature, whereas other states can yield work from sufficiently many copies. The resulting Thermal Operations preserve the thermal state and impose energy-conservation and coherence constraints, while extensions introduce explicit scope trade-offs.

  • A state other than the background thermal state can become active across sufficiently many copies, so preparing it for free would enable work extraction.The passivity argument treats states whose energy can be lowered by suitable unitaries as resources requiring work to prepare.
  • A common background temperature is required because thermal states at different temperatures could otherwise power a Carnot engine and yield work.The framework can treat a second temperature as a resource, but bookkeeping multiple temperatures remains an open extension.
  • The thermal state is not uniquely forced by non-triviality alone, since allowing arbitrary diagonal non-thermal states may still forbid creating energy-basis coherence.This shows that the passivity rationale and the requirement of a non-trivial theory are distinct constraints.
  • The framework is scoped by finite-dimensional systems and incomplete coverage, while strong-coupling, clock, and alternative-conservation settings require further formalisation or retain ignored fluctuations.Average-energy-conserving extensions allow transformations decreasing non-equilibrium free energy but ignore energy-source fluctuations; strong coupling and clock costs remain foundational issues.
  • Thermal Operations append a thermal bath, apply a unitary commuting with the total Hamiltonian, and trace out the bath; the system thermal state is consequently fixed.The energy-preserving condition is [U, HS + HB] = 0, and every resulting channel satisfies T(γS) = γS.
  • The theory also requires symmetry under time translations, which is equivalent to forbidding free creation of energetic coherence.Thermal fixed-point stability and the no-free-coherence principle motivate Thermal Processes or Enhanced Thermal Operations.

II. THERMODYNAMIC LAWS FOR POPULATION

The section characterises thermodynamic accessibility of populations through Gibbs-stochastic matrices, which preserve the Gibbs distribution. Thermal Operations induce exactly these population transformations up to arbitrarily small error, subject to bath assumptions.

  • Thermodynamic laws for population: Thermodynamic laws are formulated as constraints defining a partial order over quantum states under allowed operations.The focus is on determining which transformations are possible, rather than only identifying impossible processes.
  • Gibbs-stochastic maps: A Gibbs-stochastic matrix is a stochastic transition matrix G satisfying Gg = g, where g is the Gibbs distribution.This provides an economical description of population transformations compatible with thermal equilibrium.
  • Action of Thermal Operations: Thermal Operations map populations according to Gibbs-stochastic matrices, and every Gibbs-stochastic population map can be approximated arbitrarily well by a Thermal Operation.The forward direction follows from trace preservation, positivity, and Gibbs-state preservation; the converse uses an explicit bath and energy-preserving-unitary construction.
  • Populations and coherence: Thermal Operations commute with dephasing, so population transformations can be analysed separately from the evolution of energy-basis coherence.For non-degenerate system Hamiltonians, populations are the diagonal occupations in the energy eigenbasis.
  • Bath assumptions: The construction realising arbitrary Gibbs-stochastic matrices effectively permits baths with infinite heat capacity or volume.More restricted bath models can impose additional physical constraints on the accessible transformations.

B. Ordering states: from entropy to majorisation

At infinite temperature, thermodynamic population transformations reduce to majorisation: a distribution can be mapped to another by a doubly-stochastic process exactly when its Lorenz curve lies above the other’s.

  • Infinite-temperature ordering: The infinite-temperature problem asks when a stochastic matrix B maps p to p′ while preserving the uniform distribution.Such matrices are doubly-stochastic, or bistochastic, and represent mixing processes.
  • Majorisation: Majorisation is a partial ordering defined by comparing cumulative sums of vectors after sorting their entries in non-increasing order.It is not a total order: some distributions cannot be ranked against one another.
  • Lorenz curves: A distribution x majorises y exactly when the Lorenz curve of x lies everywhere above that of y and both curves end at the same height.Intersecting Lorenz curves indicate that neither distribution majorises the other.
  • Central theorem: The Hardy–Littlewood–Polya theorem states that x majorises y if and only if a doubly-stochastic matrix maps x to y.Thus majorisation gives the exact criterion for mixing processes with the uniform distribution as a fixed point.

2. Entropies vs majorisation

Majorisation captures a finer ordering than any single entropy, while thermo-majorisation extends this ordering to finite temperature and nontrivial Hamiltonians. The resulting criterion exactly characterises approximate population transformations under Thermal Operations.

  • Entropies vs majorisation: Schur-concave functions, including Shannon and Rényi entropies, are monotones under doubly-stochastic maps but capture only aspects of majorisation.They preserve the partial-order structure rather than fully specifying it individually.
  • Entropies vs majorisation: H(x) ≤ H(y) is necessary but not sufficient for x ≻ y, because distributions can have ordered entropies while remaining incomparable under majorisation.The example z and y demonstrates that no uniform-preserving stochastic process maps z into y despite H(z) < H(y).
  • Asymptotic conversion: For many-copy approximate conversion, the maximum ratio M/N equals (log n − H(x))/(log n − H(y)), and Shannon entropy becomes the unique asymptotic monotone.Typicality arguments explain why large tensor powers concentrate near sets whose sizes scale exponentially with Shannon entropy.
  • Thermo-majorisation: The embedding Γd replaces each probability xi by di copies of xi/di, mapping the Gibbs distribution to a uniform distribution in a larger space.This creates a bridge from Gibbs-preserving transformations to ordinary majorisation.
  • Thermo-majorisation: A Gibbs-stochastic map G satisfying Gx = y exists if and only if Γd(x) majorises Γd(y).The embedding criterion is equivalent to thermo-majorisation, defined by β-ordering entries according to xi/gi and comparing the resulting curves.
  • Thermal Nielsen theorem: Thermal Operations can transform a population x arbitrarily close to y if and only if x thermo-majorises y.This fully solves interconversion for states commuting with the Hamiltonian; coherence requires additional symmetry constraints.

2. Free energy vs thermo-majorisation, ‘second laws’ and catalysis

Thermo-majorisation refines free-energy monotonicity by characterising state transformations more strictly. A family of Rényi-based free energies becomes sufficient for diagonal transformations when catalysts are allowed.

  • Thermo-majorisation and free energy: Thermo-majorisation is preserved by thermodynamic Schur-concave and Schur-convex functions, which capture aspects of its partial ordering.These functions generalise Schur-concavity and convexity from standard majorisation.
  • Thermo-majorisation and free energy: Free energy decreases under Gibbs-stochastic maps, but its decrease alone does not guarantee a physical transformation.The example has F(x) ≈ 0.084 > F(y) ≈ −0.197 while the thermo-majorisation curves cross.
  • Many second laws: The α-free energies Fα monotonically decrease under Gibbs-stochastic maps, with F1(x) equal to the usual free energy F(x).For all α, Fα(g) = −kT log ZS.
  • Catalysis and sufficiency: All Fα inequalities together with the Burg free-energy condition are sufficient for diagonal transformations when catalysts are allowed.The theorem requires full-support, distinct distributions and strict inequalities for every α ∈ R\{0} plus the Burg condition.
  • Catalysis and sufficiency: Catalytic thermo-majorisation is equivalent to ordinary majorisation after embedding populations and tensoring with a catalyst state.The proof uses the Klimesh–Turgut result and the bridge between embedded and thermo-majorisation orderings.
  • Asymptotic regime: In the asymptotic limit, the optimal transformation rate is R = (kT log Z + F(x))/(kT log Z + F(y)).The rate applies to many uncorrelated or weakly correlated particles under Thermal Operations with negligible error.

3. Application: work extraction and work of formation for incoherent states

For incoherent states, thermo-majorisation curves reduce deterministic work extraction and formation to geometric rescaling and curve-ordering problems. These tasks are generally irreversible because formation requires more work than deterministic extraction can recover.

  • Deterministic work extraction: Deterministic work extraction maximises W while the excited-battery thermal curve remains entirely below the initial-state curve.The optimal value Wdet is fixed when the thermal curve’s elbow lies on the initial curve.
  • Curve rescaling: A battery excitation compresses a system’s thermo-majorisation curve along the x-axis by e−βW = gW.The y-axis coordinates remain unchanged, while the x-axis is rescaled.
  • Deterministic work extraction: No deterministic work can be extracted from states with full support, while average extractable work satisfies Wave = kTS1(x∥g) > Wdet.Allowing an ε probability of failure leads to extensions beyond deterministic extraction.
  • Work of formation: Work of formation is the minimum battery investment needed to transform the thermal state into a target state under Thermal Operations.For diagonal targets, the condition reduces to making the thermal curve’s slope exceed the target curve’s largest slope.
  • Irreversibility: Because S0(x) < S∞(x) for every non-thermal distribution, creating x requires more work than can later be deterministically extracted from it.Thus the cycle g → x → g is irreversible.

III. THERMODYNAMIC LAWS FOR COHERENCE

Thermo-majorisation constrains population dynamics but does not fully characterize quantum-state transformations involving coherence. Additional relations are needed to constrain how coherent amplitudes can change.

  • Population and coherence: Thermal Operations can drive populations toward the Gibbs distribution while independently degrading superpositions of energy eigenstates.For |ψ⟩ = (|0⟩ + |1⟩)/√2, the initial coherence amplitude is |c| = 1/2.
  • Population and coherence: Given a diagonal transition x → y, the theory seeks explicit constraints on the achievable final coherence amplitude |c′|.This motivates extending population-based thermo-majorisation constraints.
  • Beyond thermo-majorisation: Thermo-majorisation together with positivity is insufficient to determine all quantum transformations.The thermal state and a coherent state can share the same population vector while only the former is reachable from the thermal state.

A. Time-translation symmetry and thermodynamics

Time-translation symmetry provides the framework for thermodynamic constraints on quantum coherence in open systems. Covariant operations cannot increase asymmetry, and asymmetry monotones can rule out transformations that conserve symmetry generators.

  • Symmetry and coherence: A G-covariant channel commutes with the group action and defines the free operations of a resource theory of coherence between generator eigenspaces.For Hamiltonian-generated U(1), these are time-translation-symmetric, phase-covariant channels.
  • Symmetry and coherence: A state is symmetric when it is invariant under every group action, containing only incoherent mixtures of distinct generator eigenspaces.For time translations generated by the Hamiltonian, this corresponds to energy-basis coherence constraints.
  • Asymmetry monotones: Asymmetry monotones cannot increase under symmetric operations, replacing conservation laws for open systems.The monotonicity follows from relative-entropy contractivity and covariance.
  • Closed-system limitations: Conservation of all symmetry generators can still fail to characterize transformations under closed symmetric dynamics.The example preserves the reduced first-system state and all generator expectation values but admits no symmetric transformation.
  • Closed-system limitations: The Holevo asymmetry monotone distinguishes the forbidden transformation by giving Ap(ξSA) = log 2 while Ap(ρSA) = 0.Since asymmetry would increase, no symmetric dynamics can map ρSA into ξSA.
  • Open-system dilation: Time-translation-covariant channels admit dilations with a stationary ancilla and an energy-conserving joint unitary.The dilation connects covariance of open dynamics with conservation laws on an enlarged system.

2. Time-translation symmetry of Thermal Operations

Thermal Operations are time-translation symmetric, so they cannot freely supply coherence and must monotonically reduce asymmetry measures. These symmetry constraints supplement thermo-majorisation and are not fully captured by free-energy conditions.

  • Time-translation symmetry: Thermal Operations commute with time translations, making them U(1)-covariant and preventing external coherence from entering freely.This symmetry implies that applying the operation before or after free evolution gives the same final state.
  • Coherence monotones: Because symmetric evolutions degrade asymmetry, measures such as A(ρS) and quantum Fisher information cannot increase under Thermal Operations.For quantum Fisher information, contractivity under quantum channels establishes Q(T(ρS),t) ≤ Q(ρS,t).
  • Operational constraints: Thermal Operations satisfy Gibbs preservation and time-translation symmetry, respectively accounting for work resources and coherent resources.The two conditions are T(γS)=γS and T ◦ Ut = Ut ◦ T for all t.
  • Open questions: The relationship between Thermal Operations and Thermal Processes remains conjectural beyond the cases where their reachable transformations are known to coincide.The closure of states achievable by Thermal Operations is conjectured to equal the states achievable by Thermal Processes.
  • Limits of the constraints: Thermo-majorisation and asymmetry constraints are necessary but not sufficient to characterise Thermal Operations.The asymmetry inequalities were proved for the larger class of time-translation-symmetric channels, while Thermal Operations additionally act independently on coherence modes.

3. Coherence constraints are not reducible to free energies. Coherent and incoherent components of the free energy

Time-translation symmetry imposes coherence constraints that total or generalized free energies cannot capture. The quantum free energy separates into independently non-increasing incoherent and coherent components, producing work-locking effects.

  • Beyond free energies: Free-energy monotones alone cannot rule out transformations that increase coherence, so additional symmetry-based constraints are required.For sufficiently large E, all free-energy constraints can permit |E⟩→σS^ϵ even though the transition remains impossible.
  • Free-energy decomposition: Both the classical free-energy component and the asymmetry component independently decrease under Thermal Operations.This follows from symmetry, commutation with dephasing, and contractivity of relative entropy.
  • Free-energy decomposition: The quantum free energy additively decomposes into classical free energy of populations and a coherent component kT A(ρS).The classical term measures distance from a thermal population, while A(ρS) measures distance from the closest incoherent state.
  • Work-locking: The transition |E⟩→|+⟩ is impossible because its coherent component would increase from A(|E⟩)=0 to A(|+⟩)=log 2.This remains forbidden even when the classical and total quantum free energies decrease for sufficiently large E.
  • Work-locking: Work locking prevents the coherent part of a state's free energy from being converted into work under Thermal Operations.Work extraction from a coherent state cannot exceed extraction from its dephased state, despite ΔF(ρS)>ΔF(D(ρS)).
  • Coherence resources: Accessing coherent free energy requires an external ancillary coherence source that breaks time-translation symmetry on the work-storage system.With a very large but finite coherent source, the standard ΔF(ρS) work value can be approached, but back-reaction and implementation constraints remain.

5. Modes of coherence and hierarchy of thermodynamic constraints

Quantum states decompose into coherence modes associated with Bohr frequencies, and time-translation covariance constrains each mode separately. Thermo-majorisation is the zero-frequency member of this hierarchy.

  • Modes of coherence: A quantum state decomposes into modes of coherence, each transforming under time translations with a phase determined by its Bohr frequency.Each mode ρS^(ω) satisfies Ut(ρS^(ω)) = e^-iωt ρS^(ω).
  • Modes of coherence: The Bohr spectrum consists of energy differences between pairs of system energy eigenvalues.These frequencies label the coherence modes in the energy eigenbasis.
  • Covariance: Time-translation covariance is equivalent to preserving the mode structure under the channel.The modes can be constructed systematically using an irreducible tensor-operator basis.
  • Hierarchy of constraints: Nonzero coherence modes add further thermodynamic constraints beyond thermo-majorisation.The resulting hierarchy separates population constraints from constraints on coherences associated with distinct transition frequencies.
  • Hierarchy of constraints: Thermo-majorisation constrains the zero mode, which corresponds to the vector of energy-level populations.The zero-mode condition is equivalent to the existence of a Gibbs-stochastic matrix mapping the initial populations to the final ones.

B. Thermodynamic constraints on the evolution of quantum coherence

The section develops symmetry-based constraints on quantum coherence alongside population dynamics, then applies them to characterize achievable transformations for a single qubit under Thermal Operations.

  • A general theorem connecting population and coherence constraints: Complete positivity constrains coherent evolutions once a channel’s population dynamics is fixed.The constraint is analyzed through the positivity of the channel’s Choi matrix.
  • A general theorem connecting population and coherence constraints: Time-translation symmetry makes the Choi matrix block diagonal, with each block labeled by an energy difference ω.Positivity can therefore be checked block by block.
  • A general theorem connecting population and coherence constraints: Within each ω block, diagonal elements encode transition probabilities while off-diagonal elements encode transition amplitudes.This structure links population transitions to coherence constraints in covariant channels.
  • Application: qubit Thermal Operations: The section presents the complete solution of Thermal Operations for qubits and connects it to the geometry of reachable regions on the Bloch sphere.The regions are represented in the xz plane because of rotational symmetry about z.
  • Application: qubit Thermal Operations: For a single qubit, Gibbs-stochastic population dynamics reduce to a single parameter λ, and the achievable coherence boundary is attained by a Gibbs-preserving, time-translation symmetric channel.States inside the boundary can be obtained by following the optimal channel with partial dephasing.

3. Application: irreversibility in coherence transfers

The section studies how Thermal Operations transport coherence between equally spaced energy levels. It finds an asymmetric behavior: upward transfer suffers exponential loss, whereas the reverse transfer can be perfect.

  • Application: irreversibility in coherence transfers: Upward coherence transfer from levels 0–1 to levels 1–2 can be performed, but an exponential amount of coherence is lost.The relevant target is the largest achievable final coherence |σ12|, without fixing the final population.
  • Application: irreversibility in coherence transfers: The bound on upward transfer is achievable using a single bosonic mode prepared in a thermal state and an energy-preserving unitary.The construction supplies an explicit realization of the bound.
  • Application: irreversibility in coherence transfers: The reverse coherence transfer from levels 1–2 to levels 0–1 can be performed perfectly.This contrasts directly with the loss incurred when coherence is transported upward in energy.
  • Application: irreversibility in coherence transfers: The example shows that irreversibility of energy transfers under Thermal Operations is reflected in irreversibility of coherence transfers within a mode.The population constraints enter through Gibbs-preserving conditions on the transition matrix.

APPENDIX: PROOF OF HARDY-LITTLEWOOD-POLYA THEOREM (THEOREM 3)

The appendix proves the Hardy–Littlewood–Pólya characterization of majorization by showing that majorization is equivalent to doubly-stochastic transformations and their constructive decompositions.

  • APPENDIX: PROOF OF HARDY-LITTLEWOOD-PÓLYA THEOREM (THEOREM 3): For vectors in R^2, majorization is equivalent to obtaining one vector from the other through a doubly-stochastic matrix.The proof then proceeds inductively for higher dimensions.
  • APPENDIX: PROOF OF HARDY-LITTLEWOOD-PÓLYA THEOREM (THEOREM 3): Composing the T-transforms yields a convex combination of permutations and therefore a doubly-stochastic matrix.This establishes the constructive direction of the theorem.
  • APPENDIX: PROOF OF HARDY-LITTLEWOOD-PÓLYA THEOREM (THEOREM 3): Conversely, a doubly-stochastic matrix preserves the majorization inequalities, implying x ≻ y.The argument assumes the components of x are sorted in non-increasing order.
  • APPENDIX: PROOF OF HARDY-LITTLEWOOD-PÓLYA THEOREM (THEOREM 3): If x ≻ y, then y lies in the convex hull of the permutations of x.This is one of the equivalent conditions derived from the majorization relation.
  • APPENDIX: PROOF OF HARDY-LITTLEWOOD-PÓLYA THEOREM (THEOREM 3): Any majorized vector y can be reached through a sequence of T-transforms, each acting nontrivially on a two-level subsystem.The construction mixes the largest component with a selected component before applying the induction hypothesis.
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