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The second laws of quantum thermodynamics

Fernando G. S. L. Brandao, Michał Horodecki, Nelly Huei Ying Ng, Jonathan Oppenheim, Stephanie Wehner

arXiv:1305.5278v4quant-phcond-mat.stat-mechcs.IT

TL;DR

The paper asks whether thermodynamics has a second law for microscopic systems and develops a framework for such transformations. It finds that generalized free energies form a family of nonincreasing constraints, alongside energy conservation, with thermal states uniquely preserving a nontrivial theory.

  • Problem

    The paper investigates whether a second law constrains state transformations in microscopic systems interacting with a heat bath.

  • Method

    The paper characterizes thermodynamic transformations using energy-conserving thermal operations, generalized free energies based on Rényi divergences, and thermal-state conditions for nontriviality.

  • Results

    For a single heat bath, every Fα(ρ, ρβ) with α ≥0 cannot increase, and satisfying all these inequalities is sufficient for catalytic thermal transformation.

  • Takeaways & Limitations

    Microscopic thermodynamics is governed by a family of second laws rather than one free-energy constraint, combining monotonicity with energy conservation.

  • Takeaways & Limitations

    For fully quantum states, the stated second-law conditions are necessary but not sufficient for thermal operations.

Abstract

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The second law of thermodynamics tells us which state transformations are so statistically unlikely that they are effectively forbidden. Its original formulation, due to Clausius, states that "Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time". The second law applies to systems composed of many particles interacting; however, we are seeing that one can make sense of thermodynamics in the regime where we only have a small number of particles interacting with a heat bath. Is there a second law of thermodynamics in this regime? Here, we find that for processes which are cyclic or very close to cyclic, the second law for microscopic systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. In particular, we find a family of free energies which generalise the traditional one, and show that they can never increase. We further find that there are three regimes which determine which family of second laws govern state transitions, depending on how cyclic the process is. In one regime one can cause an apparent violation of the usual second law, through a process of embezzling work from a large system which remains arbitrarily close to its original state. These second laws are not only relevant for small systems, but also apply to individual macroscopic systems interacting via long-range interactions, which only satisfy the ordinary second law on average. By making precise the definition of thermal operations, the laws of thermodynamics take on a simple form with the first law defining the class of thermal operations, the zeroeth law emerging as a unique condition ensuring the theory is nontrivial, and the remaining laws being a monotonicity property of our generalised free energies.

A family of second laws

For cyclic thermodynamic processes with catalysts returned to their initial state, state transitions obey a family of free-energy monotones rather than a single second-law constraint. These conditions extend to fully quantum states, while the ordinary Helmholtz free energy emerges as the α→1 limit.

  • A family of second laws: The transition laws form a family of free energies Fα rather than a single free-energy condition, with necessary and sufficient conditions for the considered transformations.The family is indexed continuously by α.
  • A family of second laws: For every α≥0, Fα(ρ,ρβ)≥Fα(ρ′,ρβ), and satisfying all these inequalities is sufficient for a catalytic thermal operation from ρ to ρ′.The free energies are monotones: the system moves closer to the thermal state.
  • Macroscopic limit: For α→1, Fα becomes the ordinary Helmholtz free energy, while in macroscopic weakly correlated systems the family approximately coincides with the usual single constraint.Thus the standard second law is one member of the broader family.
  • Catalysis and work extraction: Catalytic ancillas can enable transitions that violate thermo-majorisation and can increase extractable work during cyclic memory resets.The latter includes resetting a memory while retaining correlations with a reference system.
  • Quantum second laws: For non-energy-diagonal states, quantum α-free energies give necessary but not sufficient conditions because thermal-state-preserving operations can allow transformations unavailable to thermal operations.The quantum generalization uses quantum Rényi divergences.

Work distance

The paper defines a work distance from the free-energy differences between initial and final states and uses it to characterize deterministic work extraction or investment.

  • Work distance: A work bit, or wit, changes between energy eigenstates to represent deterministic work extraction when its final energy is higher or work investment when it is lower.Its energy change supplies or absorbs the work needed for the state transformation.
  • Work distance: The maximum achievable deterministic work is W=Dwork(ρ≻ρ′), defined by the infimum over α of the free-energy difference between the initial and final states.The transition conditions imply that this value is achievable.
  • Work distance: The work distance is the minimal free-energy difference and acts as a distance measure between states, giving the operational work interpretation of the generalized free energies.This parallels the role previously assigned to thermo-majorisation criteria.

Approximately cyclic processes

The paper shows that the applicable second laws depend on how closely an ancillary working body is returned to its initial state. Exact or suitably controlled approximate cycles recover generalized second laws, whereas unrestricted small-error catalysis permits arbitrary transformations.

  • Approximately cyclic processes: Catalytic thermal operations borrow an ancillary working body, require it to return close to its initial state, conserve total energy, and discard the bath and catalyst afterward.The allowed closeness determines which second-law family applies.
  • Approximately cyclic processes: The form of the second law for inexact catalysis depends on the degree of closeness between the catalyst’s initial and final states.The paper identifies three regimes of approximate cyclicity.
  • Approximately cyclic processes: When the working body changes by at most a small work distance, the generalized second laws are recovered.In this regime, changes in the working body can be corrected using a small amount of work.
  • Approximately cyclic processes: When the catalyst’s error is at most ϵ/log N, the ordinary free energy governs transitions while the Rényi-divergences do not.Here N is the catalyst dimension, and the standard second law also arises in the macroscopic limit.
  • Approximately cyclic processes: For fixed error ϵ independent of catalyst size, arbitrarily small error can still permit heat transfer from cold to hot and arbitrary state transformations.The construction uses a sufficiently large working body, producing an apparent Clausius-law violation.

Discussion

The discussion recasts thermodynamics as a resource theory: energy conservation defines allowed operations, thermal states ensure nontriviality, and generalized free energies constrain transitions. These information-theoretic laws apply beyond microscopic systems and do not require restricted experimental control.

  • Discussion: The second-law limitations remain valid even when an experimenter can access and arbitrarily manipulate the heat bath’s microscopic degrees of freedom.Resetting the information acquired by such control still requires work, as in Maxwell’s-demon reasoning.
  • Discussion: The paper derives a family of generalized-free-energy limitations for both quasiclassical and fully quantum state transformations.These limitations combine with energy conservation to form the paper’s second laws.
  • Discussion: Catalytic thermal operations consist of borrowing a catalyst, adding copies of a state, applying an energy-conserving unitary, and tracing out the reservoir and catalyst.The catalyst is returned within a specified closeness tolerance.
  • Discussion: Energy conservation is treated as the first law and defines the allowed thermodynamic operations through unitaries commuting with the total Hamiltonian.The paper views the first law as defining thermodynamics rather than deriving from it.
  • Discussion: The zeroeth law requires free states to be thermal, because any other freely replicable state makes arbitrary transitions possible and trivializes the theory.This condition is connected to complete passivity of thermal states.
  • Discussion: Monotonicity of generalized free energies relative to the thermal state provides information-theoretic criteria for possible quantum and long-range-interaction thermodynamic transitions.The derivation requires neither ergodicity, mixing, coarse-graining, nor lack of control.

I. SUPPLEMENTARY INFORMATION

The supplementary material develops the mathematical tools underlying the paper’s transition criteria, including catalytic transformations, Rényi divergences, quantum extensions, and majorization. It also summarizes how different catalyst-return conditions produce different second laws.

  • I. SUPPLEMENTARY INFORMATION: The supplement introduces majorization, trumping, and their relation to Rényi divergences as foundations for catalytic transformation criteria.These tools reduce the number of relations needed in the bistochastic setting.
  • I. SUPPLEMENTARY INFORMATION: Different regimes of catalyst closeness yield different second laws, ranging from exact catalysis to arbitrary state transformations under sufficiently permissive return conditions.The regimes include small work correction, small error per particle, average work correction, and arbitrarily high fidelity.
  • I. SUPPLEMENTARY INFORMATION: Rényi divergences are defined over α∈[−∞,∞], with limiting conventions for α=0, 1, ±∞ and extensions to negative α.The supplement also records relations between divergences and their monotonicity properties.
  • I. SUPPLEMENTARY INFORMATION: The relative Rényi divergence satisfies the data-processing inequality for all α∈[−∞,∞].For α∈[0,∞], the Rényi divergence is additionally nondecreasing in α.
  • I. SUPPLEMENTARY INFORMATION: The supplement generalizes Rényi divergences to quantum states, where noncommutativity produces multiple candidate definitions with different monotonicity ranges.One version is monotonic under quantum operations for α∈[0,2], while another is monotonic for α≥1/2.
  • I. SUPPLEMENTARY INFORMATION: Rényi entropies are strictly Schur concave for finite nonzero α, while their limiting cases are Schur concave.The result follows from majorization and the convexity criterion for symmetric sums.

Appendix B: Exact catalysis with trivial Hamiltonian

With trivial Hamiltonian, ordinary state transitions are governed by majorization, while exact catalysis enlarges the allowed transitions and is characterized by trumping conditions involving Rényi quantities.

  • Noisy operations: With maximally mixed states and noisy operations, a transition is possible exactly when the input spectrum majorizes the output spectrum.This describes the non-catalytic resource theory with trivial Hamiltonian.
  • Catalytic transitions: Catalytic transitions are governed by trumping: some transitions forbidden by majorization become possible when an auxiliary state is returned unchanged.The catalyst enables transitions of the form x ⊗ z ≻ y ⊗ z.
  • Trumping conditions: Klimesh–Turgut conditions give necessary and sufficient trumping criteria using functions related to Rényi entropies or divergences.The conditions are expressed for all α ∈ (−∞, ∞), under the stated nonzero-component assumptions.
  • Trumping conditions: Non-strict inequalities suffice for exact trumping, allowing discrete conditions to be recovered as limits of the remaining conditions.The result uses continuity of majorization conditions and continuous functions of ordered probability distributions.
  • Approximate catalysis: Negative-α conditions are sensitive to zero components, because truncation and divergence behavior can change discontinuously under small perturbations.This instability is specific to the negative-α functions; positive-α divergences do not depend on additional zeros.
  • Approximate catalysis: When approximate output or arbitrarily small work investment is allowed, only positive-α conditions are relevant, although the construction may require borrowing a perfectly pure state.In thermodynamic terms, borrowing a perfectly pure state corresponds to initially investing an infinite amount of work.

Appendix D: Equivalence relations from set of allowed resources defines the zeroeth law

The zeroeth law identifies Gibbs states as the unique freely available resource that preserves nontrivial state-conversion conditions. The appendix establishes this through work extraction from non-Gibbs resources and develops d-majorization as the general transition framework.

  • Zeroeth law: Gibbs states are the unique free resource that does not make thermal state-conversion conditions trivial.Allowing arbitrarily many copies of any other state permits deterministic work extraction in the framework described.
  • Complete passivity: Complete passivity means that every tensor power remains passive; in finite-dimensional systems, the completely passive states are Gibbs states or ground states.Passivity requires energy-basis diagonality and populations that do not increase with energy.
  • Work extraction: A non-passive diagonal state yields a nonzero amount of extracted work from sufficiently many copies, except with arbitrarily small failure probability.The proof combines a single-copy energy-preserving battery interaction with typicality and a concentration bound.
  • Work extraction: For a single non-passive state, population inversion extracts average work ℏω(pi − pj) > 0 from the identified energy-level pair.The battery gains ℏω with probability pi and loses ℏω with probability pj.
  • Complete passivity: Any state that is neither a Gibbs state nor the corresponding ground state becomes work-extractable from sufficiently many copies, except with probability at most ε.Such states are not completely passive, so some tensor power is non-passive and Theorem 8 applies.
  • General Hamiltonians: For general Hamiltonians, d-majorization and its catalytic extension provide the mathematical criteria for thermodynamic state transformations.The construction also establishes when catalysts may be taken diagonal for initially energy-block-diagonal states.

3. Catalytic d-majorization

Catalytic transformations are characterized by monotonicity of all Rényi divergences, yielding a family of second-law constraints rather than a single criterion. For block-diagonal states, these conditions are necessary and sufficient, while borrowing certain ancillas can remove parts of the family at a cost.

  • Catalytic d-majorization: Monotonicity of Rényi divergences provides an operational characterization of catalytic transformations.The result generalizes trumping relations and gives Rényi divergences an operational interpretation.
  • Catalytic d-majorization: For all α ∈(−∞, ∞), Dα(p||q) ≥Dα(p′||q′) is equivalent to the existence of approximate catalytic transformations under the stated conditions.The equivalence is formulated for full-rank q and q′, with arbitrarily small approximation error in the catalytic construction.
  • Catalytic d-majorization: A catalyst can be chosen as a full-rank distribution, and in particular as a uniform distribution on its support.The construction shows that zeros in a catalyst do not affect the existence of the required bistochastic map.
  • Catalytic d-majorization: For diagonal system inputs, catalyst coherences do not affect the output diagonal, so a catalyst with the same diagonal and no coherences suffices.The conditional probabilities defining the induced channel depend only on the catalyst’s diagonal.
  • Catalytic d-majorization: For block-diagonal energy states, catalytic thermal transformations are possible with arbitrary accuracy if and only if the full family of α-indexed inequalities holds.The theorem applies to all α ∈(−∞, ∞).
  • Catalytic d-majorization: Borrowing a trivial-Hamiltonian pure qubit removes the α<0 restrictions, while a high-energy ancilla can make Dα arbitrarily large for any fixed α0>1.The latter ancilla is expensive because it requires substantial work to create and is returned as a thermal state.

4. Quantum second laws: limitations for states that are not diagonal in energy basis

For non-energy-diagonal states, the paper derives quantum Rényi-free-energy constraints as necessary conditions for transformations. These quantum second laws are not sufficient, and their applicability depends on how catalysts are returned and on the allowed Rényi orders.

  • Quantum limitations: Quantum alpha-free energies generalize the classical limitations for states that are not diagonal in the energy basis.They are based on quantum Rényi divergences and form a family of fully quantum second laws.
  • Quantum limitations: Quantum limitations are derived using monotonicity and additivity of quantum Rényi divergences under completely positive trace-preserving maps.The stated restriction to selected α ranges reflects divergence issues for non-full-rank catalysts.
  • Quantum limitations: The quantum second laws are necessary but not sufficient conditions for state transformations.Operations preserving the thermal state can satisfy these laws while producing transitions that thermal operations cannot perform.
  • Quantum Maxwell demon: The Maxwell-demon application considers resetting a system while leaving the demon’s correlated memory unchanged.The reset cost is expressed through the work distance associated with the generalized free energies.
  • Quantum Maxwell demon: A negative reset quantity means the demon gains work while resetting the system to a pure state.Catalytic operations can yield more work than would otherwise be possible, giving Rényi-entropy differences an operational interpretation.

2. The embezzling state dilemma: when there is no second law

Approximate catalysis defined only by fidelity or trace distance can enable work embezzlement from increasingly large catalysts. In this regime, catalysts remain arbitrarily close to their initial states while state transformations that would otherwise be forbidden become possible.

  • Embezzling dilemma: Increasing the catalyst dimension allows arbitrary transformations with arbitrarily good fidelity while returning the catalyst arbitrarily close to its initial state.This is the embezzling phenomenon adapted from entanglement theory for a trivial Hamiltonian.
  • Embezzling dilemma: A second work-embezzling example uses a nontrivial Hamiltonian together with a pure-state coherence resource.The paper identifies this as another instance of the same phenomenon.
  • Embezzling dilemma: A catalyst shifted by the energy-conserving map can return arbitrarily close to its original state while transferring one unit of work to the system.The catalyst is a superposition over energy eigenstates, and increasing its dimension reduces the detectable change.
  • Embezzling dilemma: Trace-distance closeness is too weak because restoring the original catalyst may still require substantial work.Thus small distinguishability does not ensure a small thermodynamic cost of the catalyst’s change.
  • Embezzling dilemma: If approximate cyclicity means arbitrarily high fidelity of the returned working body, then all state transformations become possible and no second law remains.The conclusion applies to the fidelity-based notion of approximate cyclicity considered here.

3. The work distance

The work distance quantifies the work needed to restore an imperfectly returned catalyst, making inexact catalysis operationally precise. It reduces to extractable work or work cost when one endpoint is thermal.

  • The work distance measures the work required to restore a returned catalyst to its original state.A battery can provide or extract this work during the transformation.
  • The transition conditions are expressed through Rényi divergences, with the allowed restoration work bounded for every α ≥ 0.The maximal extractable work is determined by the tightest bound across positive α.
  • The work distance recovers previously known extractable-work and formation-cost quantities as catalyst-independent special cases.
  • When the final state is thermal, the work distance equals maximum extractable work; when the initial state is thermal, its negative equals minimum work cost.

4. Small error per particle – recovering the free energy

Allowing a catalyst to return with sufficiently small error per particle changes the governing constraints from the full Rényi family to the ordinary free-energy law. The construction establishes this recovery for trivial and nontrivial Hamiltonians.

  • An error per particle bounded by ϵ/log(N) allows recovery of the ordinary second law.Here N is the catalyst dimension.
  • In the extensive regime, only Shannon entropy remains relevant, while the Rényi entropies cease to constrain transitions.
  • The catalyst construction uses a dimension N = n2^(n−1) and produces catalysts for sufficiently large N.
  • For nontrivial Hamiltonians, relaxing cyclicity likewise recovers the usual second law expressed through standard free energy.The standard free energy is F = E − TS.
  • The protocol can return the catalyst with exponentially small restoration work while delivering the required output state.
  • Whether restoration by acting on the whole system and ancilla is equivalent to acting only on the catalyst remains open.

Appendix H: Proofs of properties of R´enyi entropies and divergences

The appendix studies smoothed Rényi entropies and divergences, which preserve continuity under small distribution changes and simplify approximate transformation conditions.

  • Smoothing preserves continuity under small changes in probability distributions and gives the quantities operational interpretations.

b. Technical lemmas

The technical lemmas construct nearby distributions that bound smoothed Rényi entropies and divergences, supporting the reduction of approximate transition conditions. They also establish monotonicity for negative-order Rényi entropies.

  • Technical lemmas: For 0 < α < 1 and ϵ > 0, a sub-normalized distribution within the ϵ-ball can be chosen to satisfy the required entropy bound.
  • Technical lemmas: When smoothing is impossible, the bound can become trivial under sufficiently small ϵ.
  • Technical lemmas: For α > 1, smoothing is constructed by modifying probabilities relative to q, assuming supp(p) ⊆ supp(q).
  • Technical lemmas: The resulting smoothed-divergence corollaries apply separately to α > 1 and α < 1.
  • Technical lemmas: Rényi entropies are monotonically increasing in α for α′ ≤ α < 0.

2. Smoothing relations

The paper reduces the infinite family of Rényi-entropy and Rényi-divergence conditions approximately through smoothing strategies, often allowing verification using only two endpoint conditions.

  • Approximate verification: The infinite family of conditions can often be approximately verified by checking just two conditions when only α ≥0 matters.This verification is one-directional and generally sufficient in the relevant setting.
  • Rényi entropies: For 0 < α < 1, the sufficient endpoint condition is H0(p′) − log 1/ϵ^(1−α) ≥ H0(q).Here p′ is an ϵ-smoothed distribution near p.
  • Rényi entropies: For α > 1, the sufficient endpoint condition is H∞(p) ≥ H∞(q′′) + log 1/ϵ^(α−1).The distribution q′′ is obtained through the corresponding smoothing strategy.
  • Rényi divergences: For nontrivial Hamiltonians, analogous smoothing relations apply to Rényi divergences Dα relative to a reference distribution or thermal state.The construction yields smoothed distributions that bound the divergence conditions across α ranges.

1. Thermal operations and time dependent Hamiltonians

Thermal operations model state changes through energy-preserving unitaries and ancillary systems, including a switch mechanism for changing Hamiltonians. This framework recovers standard work relations and constrains work extraction through generalized free energies.

  • Thermal operations: Thermal operations couple the system to thermal states and permit energy-preserving transformations, defining the operational framework for these transitions.The switch construction incorporates Hamiltonian changes into this fixed-Hamiltonian paradigm.
  • Time-dependent Hamiltonians: Changing a Hamiltonian can be represented by an ancillary switch whose state changes from |0⟩ to |1⟩ while the total Hamiltonian remains fixed.The transition is between ρ ⊗ |0⟩⟨0| and σ ⊗ |1⟩⟨1|, with H0 and H1 acting on the system.
  • Work extraction: For thermal initial and final states, the required deterministic work equals the difference of standard free energies.This result holds for the switch-based representation of Hamiltonian changes.
  • Work extraction: For general states, a transition with a work bit is possible if and only if the generalized free energies satisfy the corresponding monotonicity conditions.The free energies Fα are defined relative to the thermal state of the Hamiltonian.
  • Work-system universality: Reversible work-system transformations require the free-energy difference to be independent of α, defining a common amount of work.For nearly pure energy states, this reduces to the work system’s energy change.
  • Work-system universality: Landauer-based purity batteries yield the same upper bound on extractable work, with no more than kT ln(2) extractable per pure bit.The net gain of pure qubits is linked to work through Landauer erasure.
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