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Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
Matteo Lostaglio, David Jennings, Terry Rudolph
TL;DR
Free-energy relations do not fully constrain thermodynamic processes involving quantum coherence. The paper treats time-asymmetry as a resource and derives independent asymmetry constraints, finding irreversible coherence transformations and structural parallels with entanglement theory.
Problem
Free-energy relations are insufficient for the single-shot thermodynamics of correlated and coherent quantum systems.
Method
The paper applies asymmetry theory to time-translation symmetry within thermal operations, extending free-energy measures to coherence measures.
Results
For all α ≥0, free-coherence changes satisfy ∆Aα ≤0 under thermal operations, independently of free-energy relations.
Takeaways & Limitations
Coherence processing has its own thermodynamic constraints and reveals structural parallels between thermodynamics and entanglement theory.
Abstract
from arXiv · showhide
Recent studies have developed fundamental limitations on nanoscale thermodynamics, in terms of a set of independent free energy relations. Here we show that free energy relations cannot properly describe quantum coherence in thermodynamic processes. By casting time-asymmetry as a quantifiable, fundamental resource of a quantum state we arrive at an additional, independent set of thermodynamic constraints that naturally extend the existing ones. These asymmetry relations reveal that the traditional Szilard engine argument does not extend automatically to quantum coherences, but instead only relational coherences in a multipartite scenario can contribute to thermodynamic work. We find that coherence transformations are always irreversible. Our results also reveal additional structural parallels between thermodynamics and the theory of entanglement.
I. RESULTS
Quantum thermodynamics requires two independent resources: thermodynamic purity, constrained by free energies, and time-asymmetry, constrained by coherence measures. These additional asymmetry laws restrict coherence processing beyond the free-energy relations.
- I. RESULTS: Thermal operations are energy-preserving interactions with a Gibbsian bath, optionally assisted by catalytic auxiliary systems.The catalyst begins and ends in the same state while enabling otherwise forbidden transformations.
- I. RESULTS: Free-energy relations are insufficient for single-shot transformations of correlated or coherent quantum states.They characterize catalytic interconversions when coherence is absent, but additional conditions arise when time-translation invariance breaks down.
- I. RESULTS: Thermodynamic operations form a strict subset of quantum operations symmetric under time translations.Consequently, thermodynamic processes cannot generate additional time-translation asymmetry.
- I. RESULTS: Quantum thermodynamics is governed by thermodynamic purity and time-asymmetry, respectively measuring thermal deviation and violation of time-translation invariance.Free energies quantify purity, while asymmetry measures quantify coherence relative to the energy basis.
- I. RESULTS: For every α ≥0, free-coherence changes satisfy ∆Aα ≤0 under thermal operations.The same constraints apply to catalytic thermal operations with energy-diagonal catalysts and extend to time-dependent Hamiltonians.
- I. RESULTS: At α →1, quantum free energy separates into classical free energy and coherence, and both contributions independently decrease under thermodynamic processes.Here A(ρ) measures coherence while Fc(ρ) is the free energy of the dephased state DH(ρ).
E. The incompleteness of existing second laws
The free-energy relations are independent of additional asymmetry constraints, so satisfying all free-energy conditions does not guarantee a thermodynamically possible coherent transformation.
- The asymmetry relations provide independent, non-trivial constraints that must hold alongside the free-energy relations for any thermodynamic transformation.
- A qubit example satisfies every free-energy condition for sufficiently small ϵ, yet remains impossible because coherence cannot be generated from a symmetric initial state.The initial state is ρ = |1⟩⟨1|, while Aα(σ) > 0 for any ϵ > 0.
- Adding sufficient work can satisfy all free-energy inequalities for arbitrary state transformations, but coherence monotones still require Aα(ρ) ≥ Aα(σ).Thus work is not a universal resource in quantum thermodynamics when coherent properties matter.
- The framework therefore treats energetic and coherent properties as jointly constrained resources rather than as aspects captured by free energy alone.
F. Emergence of classicality
Coherence constraints become negligible per particle for many non-interacting qubits, yielding an effectively classical asymptotic regime, but finite coherent systems exhibit work-locking and require relational resources for activation.
- F. Emergence of classicality: Free coherences per particle vanish for n non-interacting qubits as n →∞, producing an emergent classical scenario in which states become effectively time-symmetric.
- G. Quantum Szilard: Thermal operations yield the same work distributions for a state and its dephased counterpart, so isolated coherence cannot be extracted as mechanical work.
- F. Emergence of classicality: In the thermodynamic limit, work-locking becomes undetectable and the maximum extractable work per system approaches the quantum free energy.
- F. Emergence of classicality: For 5 qubits, up to 50% of free energy may be locked in coherence, compared with 1% for 1000 qubits.
- H. Coherent activation of work: Coherent states that cannot yield useful work alone can become useful when combined with other coherent quantum systems or an external coherent resource.The paper identifies this as coherent activation of work.
- H. Coherent activation of work: The paper draws structural parallels between quantum thermodynamics and entanglement, including irreversible formation-versus-distillation gaps and activation of otherwise unusable resources.
II. DISCUSSION
The discussion frames quantum thermodynamics through purity and time-asymmetry, extending free-energy constraints to coherent processing and identifying irreversible coherence loss and parallels with entanglement theory.
- Coherence processing has an additional irreversibility: at least A∞(ρ) − A0(ρ) coherence is lost in a cycle.
- Subsequent work used this framework to derive bounds on coherence evolution and to describe coherence through spectral mode decomposition.
- The results support structural parallels between thermodynamics and entanglement theory and point toward an explicit unification of their resource theories.
- The resource-theoretic framework describes quantum thermodynamics using two properties: thermodynamic purity and time-asymmetry.
III. METHODS
The methods define thermodynamic processes through energy-preserving interactions with a Gibbsian bath and recover traditional free-energy constraints in equilibrium cases.
- Thermal operations: Thermodynamic operations are quantum operations generated by a joint unitary on the system and environment that preserves total energy.The bath begins in a Gibbs state, and external energy sources must be included quantum-mechanically.
- Thermal operations: The bath state is γb = e^−βHb/Tr[e^−βHb], with β = (kT)−1 and total energy conserved by U.The system and bath Hamiltonians are H and Hb, respectively.
- Thermal operations: The framework permits arbitrary nonequilibrium initial and final system states and imposes no restrictions on the bath’s final state.Time-dependent Hamiltonians can be incorporated through a clock degree of freedom.
- Free-energy relations: Generalized free energies Fα provide necessary and sufficient conditions for transformations between incoherent quantum states under the considered operations.For thermal states, these conditions reduce to the equilibrium transformation criterion.
- Free-energy relations: For equilibrium-to-equilibrium transformations, the generalized conditions collapse to the traditional free-energy relation and work-extraction bound.The framework therefore agrees with the conventional thermodynamic account in this regime.
D. Symmetric operations
The paper frames symmetry and asymmetry as resources for quantum dynamics, explaining why energy measurements alone cannot capture coherence under conservation laws.
- Symmetric operations: A symmetric quantum operation respects the action of a Lie-group symmetry on the system’s Hilbert space.Symmetric states remain invariant under the corresponding unitary transformations.
- Symmetric operations: For rotational symmetry, the singlet state is invariant, whereas other states are asymmetric under the SU(2) representation.This illustrates symmetry as invariance under group transformations.
- Connection to fluctuation theorems: Fluctuation-theorem approaches capture only effectively classical stochastic effects in quantum systems.The paper contrasts this with an approach that treats coherence explicitly.
- Connection to fluctuation theorems: Moments of work distributions can be insufficient for small systems, where distributions may be broad and structured.The paper therefore points to finer-grained single-shot analyses.
- Connection to fluctuation theorems: Energy conservation is not fully captured by energy measurements because coherence properties must also be included.The work presents this as a first step toward incorporating those properties.
F. Proof of Theorem 1
The proof establishes that thermally allowed operations form a proper subset of symmetric operations.
- Proof of Theorem 1: For any bath Gibbs state, the thermal operation construction yields the required invariance using bath-state commutation and total-energy conservation.The proof uses [Hb, γb] = 0 together with [U, H + Hb] = 0.
- Proof of Theorem 1: Transforming one energy eigenstate into another is symmetric but not thermally allowed, demonstrating that the inclusion is strict.Thus symmetry alone permits operations excluded by thermal-operation constraints.
G. Thermodynamic purity
Thermodynamic transformations can be related to purity transformations through an embedding, but that construction excludes coherence and therefore requires additional relations.
- Thermodynamic purity: Time-translation asymmetry is a fundamental resource for the thermodynamics of coherent quantum states.The paper identifies thermodynamics as involving at least two independent resources: free energy and asymmetry.
- Thermodynamic purity: Earlier thermal-state interconversion results use catalysts while restricting attention to states diagonal in the energy basis.Thermal-state-preserving maps and thermal operations coincide structurally only under this incoherence restriction.
- Thermodynamic purity: The embedding Γd maps canonical distributions into a larger microcanonical space whose blocks correspond to dimensions d_i.For rational thermal distributions, the embedding maps the thermal state to a uniform distribution.
- Thermodynamic purity: Generalized free-energy differences map to thermodynamic-purity measures under Γd and its inverse.Consequently, decreasing all Fα corresponds to decreasing the associated purity measures in the embedding space.
- Thermodynamic purity: The embedded stochastic transformation can be mapped back to a thermal transformation with a catalyst of trivial Hamiltonian.The resulting map preserves the thermal state and satisfies the required transformation conditions.
- Thermodynamic purity: Because the embedding carries zero coherence deviation from equilibrium, it cannot handle coherence and must be extended beyond free-energy relations.This is the construction’s explicit limitation for coherent states.
H. Beyond free energy constraints
The paper derives coherence constraints from the symmetry of thermal operations and extends them to catalytic thermal operations with block-diagonal catalysts.
- Thermal operations are symmetric, enabling the derivation of coherence second laws from time-translation symmetry.The proof uses symmetry of the thermal operation together with an integrated relation and the data processing inequality for quantum Rényi divergences.
- The coherence second laws also hold when a catalyst block-diagonal in energy assists the transformation.This broadens the operational setting beyond ordinary thermal operations.
- A catalytic thermal operation is defined by applying a thermal operation to system and catalyst while returning the catalyst unchanged.The catalyst state is combined with the system, transformed jointly, and reproduced at the output.
- Catalytic thermal operations with block-diagonal catalysts are symmetric operations, so the coherence constraints remain applicable.The symmetry follows from the catalyst’s commutation with its Hamiltonian and the symmetry of the underlying thermal operation.
I. Coherence second laws for time-dependent Hamiltonians
The coherence second laws extend to processes with time-dependent Hamiltonians by representing the Hamiltonian switch with a classical clock degree of freedom.
- A classical clock degree of freedom represents a switch that changes the Hamiltonian during a thermodynamic process.The clock can model a control such as tuning a magnetic field.
- The resulting transformation sends a state with initial Hamiltonian H0 to a state with final Hamiltonian H1.The switch variable encodes the change from H0 to H1.
- For all α ≥0, any thermal operation switching the Hamiltonian from H0 to H1 necessarily satisfies ∆Aα ≤0.The time-dependent case follows by applying the coherence second law to the enlarged Hamiltonian containing the clock.
J. Proof of Eq. (5)
The proof of Eq. (5) reduces the asymmetry bound to pure states and bounds it using block-structured vectors and p-norm inequalities.
- The argument first reduces the proof to pure qubit states and then uses symmetry under rotations about Z to place them in the xz plane.The reduction extends the result from pure states to arbitrary states through mixing and symmetric operations.
- The proof introduces vectors whose components are grouped into blocks indexed by h to analyze n-copy states.The block structure organizes components according to the relevant excitation sectors.
- Each h-block is formed by summing equal components, allowing the vector norm to be related to the asymmetry expression.The prefactors determine the elements of the grouped blocks before substitution into the target equation.
- For α > 1, an upper bound on ||x(n)||1/α yields an upper bound on Aα, and monotonicity extends the result to all α ≥0.The p-norm identity follows from Hölder’s inequality; nonnegativity follows from properties of Sα.
K. Work-locking
The work-locking proof compares the work distributions obtainable from a state and from its energy-dephased version under time-symmetric operations.
- Energy dephasing is a time-translation symmetric operation, so any work distribution obtainable from the dephased state is also obtainable from the original state.The dephased state can be generated from the original without breaking time-translation symmetry.
- The converse direction assumes a time-symmetric operation E produces a work distribution from the original state.The proof then applies E to the energy-dephased state.
- Applying E to the dephased state establishes the corresponding work-locking relation through the paper’s asymmetry expressions.