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Quantum Thermodynamics: An introduction to the thermodynamics of quantum information

Sebastian Deffner, Steve Campbell

arXiv:1907.01596v1quant-phcond-mat.mes-hallcond-mat.quant-gascond-mat.stat-mech

TL;DR

Quantum thermodynamics needs a framework that faithfully describes quantum systems and their information-processing applications, because standard equilibrium concepts are not well phrased for quantum systems. The book develops this framework by reviewing thermodynamics, stochastic thermodynamics, quantum-information-based equilibrium, and quantum thermodynamic quantities. It presents an exact microscopic expression for mean nonequilibrium entropy production while noting that measurement-related informational costs require inclusion in a fully consistent framework.

  • Problem

    Standard equilibrium concepts such as canonical thermal equilibrium are not well phrased for quantum systems, motivating a quantum-information-based thermodynamic description.

  • Method

    The book develops quantum thermodynamics by reviewing classical and stochastic thermodynamics, characterizing equilibrium through quantum information, and introducing quantum work, heat, and entropy production.

  • Results

    The framework gives an exact microscopic expression for mean nonequilibrium entropy production in a driven open quantum system weakly coupled to one heat reservoir, including states arbitrarily far from equilibrium.

  • Takeaways & Limitations

    Quantum thermodynamics connects thermodynamic analysis with quantum information processing and the optimization of quantum technologies.

  • Takeaways & Limitations

    The two-time energy-measurement treatment omits the work cost of acquiring information, which a fully consistent thermodynamic framework should include.

Abstract

from arXiv · show

This book provides an introduction to the emerging field of quantum thermodynamics, with particular focus on its relation to quantum information and its implications for quantum computers and next generation quantum technologies. The text, aimed at graduate level physics students with a working knowledge of quantum mechanics and statistical physics, provides a brief overview of the development of classical thermodynamics and its quantum formulation in Chapter 1. Chapter 2 then explores typical thermodynamic settings, such as cycles and work extraction protocols, when the working material is genuinely quantum. Finally, Chapter 3 explores the thermodynamics of quantum information processing and introduces the reader to some more state-of-the-art topics in this exciting and rapidly developing research field.

About the Authors

The prologue frames thermodynamics as a universal theory motivated by practical technologies, then positions quantum thermodynamics as a framework for optimizing emerging quantum technologies. The book is designed as a concise graduate-level introduction connecting thermodynamics with quantum information.

  • Thermodynamics seeks universal descriptions of typical behavior rather than detailed microscopic predictions.
  • Practical questions about steam-engine efficiency helped motivate the origins and development of thermodynamics.
  • Quantum computing is presented as a developing technology whose operation should minimize wasted resources such as work and information.
  • Quantum thermodynamics must identify canonical variables beyond volume, temperature, and pressure and translate thermodynamic principles into optimization statements for quantum technologies.
  • The book provides a concise introduction to quantum thermodynamics and quantum information processing for readers with graduate-level statistical physics and quantum mechanics.

The principles of modern thermodynamics

Modern thermodynamics is an axiomatic, phenomenological framework for average heat and work, but its classical equilibrium formulation has important limitations for microscopic, nonequilibrium, and quantum systems.

  • Thermodynamics is an axiomatic phenomenological theory describing the average behavior of heat and work.
  • Its major shortcomings are the absence of microscopic information, inability to characterize nonequilibrium states, and inadequate direct applicability to quantum systems.
  • The book addresses these limitations by reviewing thermodynamics, extending it to stochastic thermodynamics, and formulating equilibrium and quantum thermodynamics through quantum information.

1.1 A phenomenological theory of heat and work

Phenomenological thermodynamics describes equilibrium states and transformations using macroscopic variables, while finite-time and irreversible processes motivate extensions beyond the equilibrium manifold. Endoreversible analysis shows that nonequilibrium engines can have universal efficiency relations at maximal power.

  • Thermodynamics describes work and heat through the average operation of engines and focuses on transformations between equilibrium states.
  • Equilibrium states are represented on a smooth thermodynamic manifold defined by an equation of state such as PV = NkBT.
  • Quasistatic processes remain near equilibrium and are treated as reversible paths on the thermodynamic manifold, whereas real finite-rate processes leave it.
  • The first law separates state-dependent internal energy changes from path-dependent work and heat exchanges.
  • The Curzon-Ahlborn result illustrates that nonequilibrium thermodynamics can yield universal, mathematically simple behavior, including efficiency at maximal power depending only on reservoir temperatures.

1.2 The advent of Stochastic Thermodynamics

Stochastic thermodynamics extends thermodynamic reasoning to small systems far from equilibrium, where fluctuations affect heat, work, and entropy production. Langevin and distribution-based descriptions support microscopic work definitions and fluctuation relations such as Jarzynski’s equality and Crooks’ theorem.

  • Thermodynamic observables fluctuate in small systems, and individual entropy or work fluctuations can violate macroscopic second-law inequalities.
  • The fluctuation theorem relates negative and positive entropy-production fluctuations, making negative events exponentially unlikely while recovering the second law on average.
  • Stochastic thermodynamics studies small systems far from equilibrium whose dynamics are governed by fluctuations.
  • The Langevin equation models a Brownian particle through deterministic forces, damping, and stochastic environmental noise.
  • The Klein-Kramers equation describes probability distributions in position and velocity space and permits direct computation of entropy production.
  • Stochastic work is defined for individual trajectories through changes driven by an external control parameter, but its fluctuations require analyzing the work distribution P(W).
  • The Jarzynski equality generalizes the second law to a broad range of classical nonequilibrium systems, while Crooks’ theorem is restricted to Markovian processes.

1.3 Foundations of statistical physics from quantum entanglement

The section addresses the difficulty of defining thermal equilibrium fully quantum mechanically and presents envariance, an entanglement-based symmetry, as the foundation for deriving equilibrium representations.

  • Motivation: Classical foundations of canonical equilibrium rely on concepts that are not well-phrased for quantum systems.Examples include Boltzmann’s H-theorem, ergodicity, ensembles, and entropy maximization.
  • Entanglement-assisted invariance: Envariance derives microcanonical and canonical equilibrium representations from symmetry considerations of entanglement.This approach replaces problematic classical foundations with a fully quantum treatment.
  • Entanglement-assisted invariance: Envariance occurs when an environment-only unitary restores the effect of a system unitary on a maximally entangled composite state.The environment performs the inverse transformation without acting directly on the system.
  • Entanglement-assisted invariance: Swapping two entangled spin states and reversing the swap in the environment leaves their local probabilities exchanged and unchanged, implying equality.The example connects quantum-state symmetries and probabilities and is presented as leading to Born’s rule.
  • Physical status: Experiments in quantum optics and on IBM’s Q Experience demonstrate that envariance is physically realizable.The text uses this experimental status to motivate envariance as a basis for quantum equilibrium states.
  • Microcanonical equilibrium: A microcanonical equilibrium is represented by an equivalence class of states that are envariant under all system unitaries and fully energetically degenerate.The defining conditions are maximal envariance and equal energy across the relevant states.

1.3. FOUNDATIONS OF STATISTICAL PHYSICS FROM QUANTUM ENTANGLEMENT 21

The section reformulates statistical-mechanical foundations using envariance and extends the construction from microcanonical equilibrium to the Boltzmann-Gibbs canonical state under thermodynamic assumptions.

  • Reformulation of the fundamental statement: Envariance replaces mathematically ambiguous notions such as probability, ergodicity, ensemble, randomness, and indifference in the quantum formulation.The reformulation uses a symmetry induced by entanglement as its central principle.
  • Microcanonical equilibrium: The quantum microcanonical equilibrium is a fully energetically degenerate state envariant under all unitaries.This is the section’s reformulated fundamental statement of statistical mechanics.
  • Canonical equilibrium assumptions: Canonical counting treats the subsystem and heat bath as non-interacting, with a sufficiently small interaction neglected through ultraweak coupling.The interaction remains physically necessary for energy exchange but is omitted from the energy and equilibrium-state calculations.
  • Canonical construction: Envariance permits the subsystem basis to be chosen as energy eigenstates, while product energy eigenstates describe the non-interacting subsystem and bath.All orthonormal bases are equivalent under envariance, enabling this energy-basis choice.
  • Canonical construction: The accessible-state count for a subsystem energy eigenstate equals the fraction determined by the total subsystem states and the bath degeneracy at the remaining energy.The bath contribution is NB(E −e_k), the number of bath states compatible with the fixed total energy.
  • Boltzmann counting: The bath’s degeneracy counts arrangements of non-interacting qubits and generalizes to Boltzmann’s counting formula for classical microstates.The quantum treatment counts degenerate states directly rather than discretizing phase space with an artificial grid.
  • Boltzmann-Gibbs state: For a very large reservoir, Stirling’s approximation and variational calculus yield the Boltzmann-Gibbs formula, with temperature entering through β.The resulting expression counts reservoir states for subsystem and bath canonical equilibrium.
  • Scope and conditions: The derivation is exact apart from Stirling’s approximation but requires additional thermodynamic conditions for the canonical state.The text contrasts these requirements with the microcanonical construction based on maximal envariance.

1.4 Work, heat, and entropy production

The section develops thermodynamic definitions for quantum systems, separating useful work, entropy-associated heat, and energetic costs of maintaining quantum correlations. It then extends these ideas to nonequilibrium dynamics, quantum work relations, and measurement-related informational costs.

  • Gibbs equilibrium states: In Gibbs states, internal-energy changes separate into useful work and the entropic contribution T dS.Work arises from changing the Hamiltonian, while heat is associated with entropy change.
  • Thermodynamics of non-Gibbsian equilibrium states: For non-Gibbsian equilibrium states, heat cannot be identified solely from changes in the system state because system–environment correlations contribute energetically.Part of the exchanged energy maintains coherence and correlations rather than changing the system’s thermodynamic entropy.
  • Thermodynamics of non-Gibbsian equilibrium states: The excess heat is the only heat contribution associated with entropic cost, while the energetic maintenance term contributes to excess work.The framework distinguishes total heat, correlation-maintenance cost, and excess work for isothermal quasistatic processes.
  • Quantum Carnot engines: The generalized framework yields a thermodynamically consistent efficiency for quantum Carnot engines, which reduces to the classical Carnot efficiency in the Gibbs limit.The cost of maintaining a non-Gibbsian equilibrium state is excluded from work that can serve external purposes.
  • Nonequilibrium quantum dynamics: For driven open quantum systems, the microscopic entropy-production expression applies to intermediate states arbitrarily far from equilibrium.The dynamics include both unitary evolution and environmental non-unitary effects, under a unique-steady-state assumption.
  • Quantum work and measurement: Two-time energy measurements support a quantum Jarzynski equality and a generalized maximum-work theorem that account for the thermodynamic cost of projective measurements.Measurement changes informational entropy even though projective energy measurements do not change internal energy.

m. As before, Πi

The section develops quantum fluctuation descriptions using sequential measurements, CPTP dynamics, and phase-space representations. It establishes when quantum efficacy becomes a fluctuation theorem and proves an integral fluctuation theorem for quantum dynamics.

  • Quantum measurement framework: Sequential measurements of observables Ωi and Ωf define possible measurement outcomes and their probability distribution after a CPTP evolution.The initial measurement can alter statistics when the state and observable do not commute; the intervening map may be unitary or non-unitary.
  • Quantum fluctuation relations: Quantum efficacy ε quantifies information lost by not measuring the environment and depends explicitly on the map E.It is therefore tied to the measurement protocol rather than being universally protocol-independent.
  • Quantum fluctuation relations: A fluctuation theorem is recovered when the initial state is proportional to an exponential state and the CPTP map is unital, E(I) = I.These conditions include initial Gibbs states, energy measurements, and unitary Schrödinger dynamics; the quantum Jarzynski equality also holds for purely decohering or dephasing models.
  • Quantum entropy production in phase space: Quantum trajectories in phase space are mathematical constructs whose physical quantities are obtained by ensemble averages.The construction defines entropy production along generalized quantum trajectories rather than asserting that individual trajectories are directly physical paths.
  • Quantum entropy production in phase space: The Wigner representation retains classical marginals and quantum information, but a Liouvillian does not generally exist for every quantum system.For a thermally open harmonic oscillator, its existence and explicit form depend on the environment’s initial preparation; stationary Wigner functions also need not represent Gibbs states.
  • Quantum entropy production in phase space: For any open or closed quantum system described by the quantum Liouville equation, the constructed entropy production fulfills an integral fluctuation theorem.The proof uses an exponentially weighted marginal of the joint phase-space and entropy-production distribution and the stationary solution of the master equation.

The advent of stochastic thermodynamics 1.2

The exercises apply classical stochastic thermodynamics to harmonic oscillators. They ask readers to derive work distributions under frequency variation or dragging and verify the Jarzynski equality or Crooks fluctuation theorem.

  • The advent of stochastic thermodynamics 1.2: For a classical harmonic oscillator with Maxwell–Boltzmann preparation, frequency variation is used to compute the work probability density and verify the Jarzynski equality.The dynamics is specified by the classical Liouville equation.
  • The advent of stochastic thermodynamics 1.2: The exercises contrast isolated frequency variation with bath-coupled dragging as two settings for analyzing stochastic work.The first uses Liouville dynamics, whereas the second uses Klein–Kramers dynamics.
  • The advent of stochastic thermodynamics 1.2: For a thermally coupled classical harmonic oscillator, dragging along the x-axis is used to compute the work probability density and verify the Crooks fluctuation theorem.The dynamics is specified by the classical Klein–Kramers equation and the initial state is Maxwell–Boltzmann.

Foundations of statistical physics from quantum entanglement 1.3

The section exercises quantum statistical foundations through entanglement symmetry and finite-size corrections. They ask readers to illustrate envariance and examine higher-order Stirling terms and temperature identification.

  • Foundations of statistical physics from quantum entanglement 1.3: The exercise asks readers to illustrate envariance for a universe consisting of two harmonic oscillators and parity-preserving unitary maps.The setup connects the concept of envariance to a concrete bipartite oscillator model.
  • Foundations of statistical physics from quantum entanglement 1.3: A second exercise repeats the derivation leading to Eq. (1.78) while including the next two terms of the Stirling expansion.The listed correction includes 2 ln(2πn) + 1/(12n).
  • Foundations of statistical physics from quantum entanglement 1.3: The exercises ask how temperature would be identified after incorporating the higher-order Stirling corrections.This question follows the finite-size expansion rather than the leading approximation alone.

Work, quantum heat, and quantum entropy production 1.4

The exercises extend work and entropy-production analysis to quantum harmonic oscillators. They compare an isolated infinitely slow protocol with a bath-coupled dragging protocol and ask readers to verify the corresponding fluctuation relation.

  • Work, quantum heat, and quantum entropy production 1.4: For a thermally isolated quantum harmonic oscillator, the exercise asks for the work probability density during an infinitely slow angular-frequency variation from a Gibbs state.The protocol is quasistatic and the initial preparation is thermal.
  • Work, quantum heat, and quantum entropy production 1.4: The bath-coupled quantum exercise uses the quantum fluctuation theorem as its verification target, whereas the isolated exercise focuses on work statistics.This separates work analysis in isolated dynamics from entropy-production analysis in open dynamics.
  • Work, quantum heat, and quantum entropy production 1.4: For a quantum harmonic oscillator coupled to a thermal bath, the exercise asks for the entropy-production probability density while dragging the oscillator along the x-axis.The dynamics follows the quantum Klein–Kramers equation and the initial state is Gibbsian.

Thermodynamics of Quantum Systems

This section develops quantum thermometry, quantum heat engines, and quantum batteries, showing how quantum systems alter precision, efficiency, work extraction, and power. It also identifies cases where quantum structure improves thermodynamic performance or changes the relevant operating conditions.

  • Quantum thermometry: Quantum thermometry uses a probe that equilibrates with the system, is decoupled, and reveals temperature through an energy measurement.The chapter introduces quantum temperature estimation before treating quantum heat engines and batteries.
  • Quantum thermometry: Smaller energy gaps improve thermometric precision, but each specified gap has a single temperature at which the QFI is maximal.Changing the energy spacing shifts the temperature of the QFI peak.
  • Quantum thermometry: For harmonic spectra, thermometric precision depends primarily on the characteristic gap ∆, while dimensionality has only a minor role.At kBT ≲0.2 with ∆=1, systems of different dimensionality perform identically; differences emerge at higher temperatures.
  • Quantum thermometry: An optimal N-dimensional thermometer maximizes equilibrium energy variance, equivalently heat capacity, and has one ground state with an (N −1)-fold degenerate excited state.The corresponding precision scales with probe dimensionality, although the passage truncates the associated qualification.
  • Quantum heat engines: The quantum TLS Otto cycle exchanges work during A →B and C →D and heat during B →C and D →A, reaching bath equilibrium only at A and C.Because the TLS is out of equilibrium with the baths during most of the cycle, the process is irreversible.
  • Quantum heat engines: TLS Otto efficiency is governed by the compression ratio κ ≡∆i/∆f and is always below Carnot efficiency under the positive-work condition.The compression-ratio dependence parallels classical engines, while the positive-work constraint limits efficiency.
  • Endoreversible Otto cycle: For a classical harmonic oscillator, the endoreversible Otto efficiency at maximal power equals the Curzon-Ahlborn efficiency.The result follows analytically for the classical harmonic oscillator.
  • Endoreversible Otto cycle: For a quantum harmonic oscillator, maximal power depends on stroke times because the efficiency expression no longer factorizes.In the classical limit ¯hωf /(kBTc) ≪1, the efficiency recovers the classical expression; in the deep quantum regime it exceeds Curzon-Ahlborn efficiency.

Thermodynamics of Quantum Information

Quantum information processing is constrained by thermodynamics because information is physically encoded, and logically irreversible erasure produces unavoidable dissipation. Landauer’s principle connects entropy reduction in a bit with heat transferred to the environment.

  • Quantum computing devices are subject to decoherence and dissipation, causing irreversible entropy production and loss of quantum information to the environment.
  • Landauer’s principle establishes an absolute minimum thermodynamic cost for erasing information, independent of other constraints.
  • Logical irreversibility occurs when an output does not uniquely determine the inputs, as in an AND gate mapping multiple inputs to one output.
  • Copying a definite bit can be thermodynamically reversible because it is one-to-one and leaves the entropy unchanged.
  • Erasing a definite bit increases its information entropy by ln2 and converts kBT ln2 of work into heat in the environment.
  • Bennett’s protocol distinguishes reversible erasure of a random bit from irreversible erasure when the initial bit state is definite.

Maxwell’s Demon and Szilard’s Engine.

The text develops quantum thermodynamic bounds and diagnostics for information processing, quantum annealing, nonequilibrium driving, and error correction. It links entropy production to system–environment correlations and finite-rate excitations, while identifying transitionless driving as a possible suppression strategy.

  • Quantum entropy production is related to correlations established between the system and its environment.
  • Landauer’s non-equilibrium quantum principle follows from non-negativity of relative entropy and mutual information, with equality only for trivial processes.
  • The quantum fluctuation theorem holds for arbitrary annealing durations when the dynamics is unitary or unital, so deviations can diagnose nonideal behavior.
  • D-Wave experiments found distributions far from the theoretical prediction and dynamics that were not even unital, with clear dependence on annealing time.
  • Departures from the ideal Ising distribution indicate kinks or topological defects, whose number measures how far the annealer misses the ground state.
  • For constant-rate driving through a critical point, excess work obeys universal scaling that depends on the driving critical exponent.
  • Quantum annealing correction was experimentally tested for n = 3 and demonstrated efficient correction of environmental noise, but not fundamentally non-correctable errors.
  • Transitionless quantum driving is presented as a way to suppress finite-time excitations, while scale-invariant protocols admit a closed auxiliary term proportional to {q, p}.

Quantum thermodynamics of information 3.1

The exercises apply quantum thermodynamic information principles to erasure in classical and quantum systems. They ask readers to verify Landauer’s principle, separate classical from quantum-correlation costs, and analyze heat dissipation.

  • A double-well exercise asks readers to verify Landauer’s principle when a Maxwell–Boltzmann state is reset into one well with accuracy δ.
  • A Bell-state exercise asks readers to calculate heat dissipated during complete erasure of stored quantum information.
  • The Bell-state exercise further separates heat due to erasing classical information from heat due to destroying quantum correlations.

Performance diagnostics of quantum annealers 3.2

The exercises use quantum thermodynamic diagnostics to analyze two-level-system dynamics, unitality, work distributions, and critical behavior. They connect master-equation properties and finite-rate driving to measurable thermodynamic quantities.

  • A weakly thermal-noisy two-level system exercise asks for the quantum efficacy of identical σx initial and final measurements from a Gibbs state.
  • A strong-coupling master-equation exercise asks readers to show that σz system–bath dynamics is unital.
  • A Landau–Zener exercise asks readers to compute the quantum work distribution for evolution from −t̂ to t̂ and show excess-work Kibble–Zurek scaling.
  • A mean-field Landau free-energy exercise asks readers to identify the critical point and exponents, then predict irreversible entropy production under finite-rate driving.

Error correction in adiabatic quantum computers 3.4

The section considers auxiliary Hamiltonians for suppressing finite-time excitations in encoded quantum annealing correction. It also asks whether the resulting total Hamiltonian could be implemented on a quantum annealer such as D-Wave.

  • The setup considers two coupled qubits described by an Ising Hamiltonian in a transverse field.
  • The auxiliary Hamiltonian H1(t) is computed for transitionless quantum driving.H1(t) is identified as Eq. (3.58).
  • For n = 2 and ⟨N⟩ = 3, the encoded Hamiltonian ⟨H(t)⟩ is used to compute the corresponding H1(t).The construction is intended to suppress finite-time excitations in quantum annealing correction.
  • The section asks whether the total Hamiltonian Htot(t) = ⟨H(t)⟩ + H1(t) can be implemented on a quantum annealer such as the D-Wave machine.
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