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The extraction of work from quantum coherence
Kamil Korzekwa, Matteo Lostaglio, Jonathan Oppenheim, David Jennings
TL;DR
The paper asks how coherence in a quantum state can be converted into work while accounting for all ancillary resources. It models bounded, reference-assisted thermal machines and analyzes their repeatable use. Coherence can be converted into work arbitrarily well with sufficiently large references, while any fixed finite machine cannot extract all available coherence work, though bounded machines can remain reusable indefinitely.
Problem
The paper focuses on work extraction from coherent quantum states and the need to account explicitly for coherence resources in thermodynamic transformations.
Method
The paper models thermal machines with coherence-carrying reference systems and batteries, analyzing individually processed systems in single-shot and asymptotic regimes.
Results
A sequence of bounded thermal machines can approach extraction of the state's free energy, while any fixed thermal machine cannot extract all available work from coherence.
Takeaways & Limitations
Coherence can enhance average and single-shot work extraction, but finite machines do not reach the classical-limit performance.
Takeaways & Limitations
Optimal repeatable coherence-to-work conversion requires an unbounded reference, while bounded references can only approach that limit.
Abstract
from arXiv · showhide
The interplay between quantum-mechanical properties, such as coherence, and classical notions, such as energy, is a subtle topic at the forefront of quantum thermodynamics. The traditional Carnot argument limits the conversion of heat to work; here we critically assess the problem of converting coherence to work. Through a careful account of all resources involved in the thermodynamic transformations within a fully quantum-mechanical treatment, we show that there exist thermal machines extracting work from coherence arbitrarily well. Such machines only need to act on individual copies of a state and can be reused. On the other hand, we show that for any thermal machine with finite resources not all the coherence of a state can be extracted as work. However, even bounded thermal machines can be reused infinitely many times in the process of work extraction from coherence.
I. INTRODUCTION
The paper examines whether quantum coherence across energy eigenspaces can be converted into work under strict energy accounting. It frames this question using thermal machines with explicitly tracked ancillary resources and reviews single-shot and average-work settings.
- Strict energy conservation prevents a reversible transformation from a superposition of energy states into an energy eigenstate.
- Quantum coherence is defined here as a superposition of states belonging to different energy eigenspaces.
- Previous individually processed approaches often use a classical external field without accounting for the thermodynamic cost of maintaining it.
- The proposed framework explicitly models coherence resources, including ancillary reference systems and battery systems that store or transfer work.
- For incoherent states, the framework assumes that average work equal to the free-energy change can be extracted in the presence of a heat bath.
- Single-shot work concerns deterministic work in one protocol instance, whereas average-work analysis concerns quantities obtained over repeated runs.
B. Work-locking
Without ancillary coherence, thermal operations commute with dephasing, so coherence cannot increase extractable work in the individually processed setting. This phenomenon is called work-locking.
- The target transformation converts a coherent system into a thermal state while increasing the free energy of an incoherent battery.
- Thermal operations commute with dephasing, which removes coherence in the energy eigenbasis.
- The extractable average work from a coherent state is bounded by the work extractable from its dephased state.
- This bound is achievable because dephasing itself is a thermal operation, producing work-locking of coherence.
- The central question is how far standard average-work and single-shot formulas extend when coherence resources are explicitly included.
C. Different thermodynamic regimes
Work extraction depends on whether systems are processed individually or collectively and whether protocols are single-shot or repeated. The paper argues that coherence can unlock additional work, but resource accounting and repeatability impose limits.
- C. Different thermodynamic regimes: Thermodynamic regimes vary by the number of systems processed per run and by whether the protocol is single-shot or many-runs.
- C. Different thermodynamic regimes: Collective processing or an ancillary quantum memory can unlock work from coherence, with sublinear reference coherence consumption in the asymptotic limit.
- 1. Average energy conservation: Average-energy-conserving approaches can extract the full free-energy difference from arbitrary quantum states, but their allowed operations depend on the processed state.
- 1. Average energy conservation: Average-energy conservation can hide arbitrarily large higher-moment energy fluctuations that are not explicitly modeled.
- 1. Average energy conservation: Strictly energy-preserving implementations must increase ancillary energy fluctuations when simulating average-energy transformations, requiring work to restore the ancilla.
- 2. Repeatable use of coherence resources: Without additional coherence resources, work-locking blocks work extraction from coherence in individually processed systems.
- 2. Repeatable use of coherence resources: Reference quality is characterized by coherence and lowest occupied energy, with the classical-reference limit reached as coherence quality approaches one.
- 2. Repeatable use of coherence resources: Reference systems supply additional coherence, while their finite resources and back-reaction must be accounted for in repeatable protocols.
III. THE PROTOCOL
The protocol targets work extraction from coherent pure qubit states by processing each copy individually with a repeatable thermal machine containing a reference. Under strictly energy-conserving operations, coherent Gibbs states cannot outperform their dephased thermal states.
- The analysis concerns work extraction from pure qubit states with coherence under individual processing and without collective operations or quantum memory.
- Repeatability requires the auxiliary reference to remain equally useful at the protocol’s end, although its state may change.
- Under strictly energy-conserving operations, coherent Gibbs states cannot yield more work than their corresponding thermal states because the states are indistinguishable.
- The thermal machine contains a reference state and implements an energy-conserving unitary to unlock work from coherence.
A. The explicit work-extraction protocol
The explicit protocol uses an energy-preserving interaction between the input system and a coherent reference, followed by work extraction, reference repumping, and repetition with a fresh input copy.
- The protocol begins by coupling the system state |ψ⟩ to the reference through an energy-preserving unitary.
- After preprocessing, work is extracted from the dephased system state and stored in the thermal machine.
- The joint interaction approximately induces a desired unitary on the system while the reference undergoes back-reaction.
- Part of the extracted work is used to repump the reference so the repeatability requirement is maintained.
- The protocol can then be repeated with the updated reference and a fresh copy of |ψ⟩.
B. Performance
Reference quality controls the induced system operation and its back-reaction. Finite references constrain complete work extraction, but increasingly large bounded references can approach the free-energy difference with near-perfect repeatability.
- The system’s final excited-state occupation q depends on the reference population R00 and quality parameter ⟨¯∆⟩.
- q approaches 1 when R00 approaches 0 and ⟨¯∆⟩ approaches 1, indicating improved ability to induce the target unitary.
- If R00 = 0 initially, preprocessing leaves ⟨¯∆⟩ unchanged.
- With repumping enforcing zero ground-state population, the reference ends with the same quality parameters and remains as useful as initially.
- Repeatability limits the reference’s average free energy because the reference must operate arbitrarily many times without changing protocol performance.
- A bounded reference cannot extract average work equal to the full free-energy difference, but a sequence of bounded references can approach that value with protocols approaching perfect repeatability.
A. Limitations of bounded thermal machines
The paper shows that bounded references cannot extract all coherence as work exactly, but sequences of bounded thermal machines can approach the ideal conversion limit arbitrarily closely.
- Limitations: The ancillary system must carry coherence because an incoherent ancilla would induce a time-translation-covariant map, unlike the target transformation.
- Limitations: A bounded reference cannot implement the required exactly unitary transformation or extract the full free-energy difference from coherence.The obstruction follows from energy preservation and the need to rotate the system’s energy eigenbasis.
- Approaching the limit: A sequence of bounded thermal machines approaches the ideal coherence-to-work conversion with arbitrarily high success probability and arbitrarily small changes in reference quality.The limiting case is reversible.
- Resource accounting: The protocol accounts for repumping and measurement costs while retaining arbitrarily exact repeatability in the appropriate limit.The measurement cost is bounded by kT h2(psucc).
- Approaching the limit: The work deficit per processed copy scales as M^-1/3 and can therefore be made arbitrarily small by increasing the number of uses M.
- Approaching the limit: The extracted work approaches ∆F(|ψ⟩) when reference quality approaches one, M grows, and M(1 − ⟨¯∆⟩) approaches zero.
V. EXTRACTING WORK WITH PERFECT REPEATABILITY AND BOUNDED THERMAL MACHINES
The paper addresses whether bounded references can support small-scale, single-shot work extraction without failure or degradation, rather than relying on unbounded or asymptotic assumptions.
- Motivation: Unbounded references and large-copy asymptotic processing may be unsuitable for microscopic or single-shot thermodynamic scenarios.
- Motivation: The paper asks whether bounded references can be reused perfectly without failure while preserving exactly the same quality parameters.
- Results: Protocols for average and single-shot extraction achieve perfect repeatability, but average work remains strictly below the free-energy difference.
- Results: Positive average-work extraction is possible only when the reference quality exceeds a threshold ⟨¯∆⟩crit.
A. Average work extraction
For the state |γ⟩, a repeatable thermal machine can unlock average work from coherence when its reference quality exceeds a threshold, with greater yields for higher quality and thermal occupation.
- Repeatability: The reference is repumped after each run so its quality parameters remain unchanged, but repumping requires an investment of work.
- Average work extraction: Positive average work requires the reference quality ⟨¯∆⟩ to exceed a threshold, and higher quality increases the average coherence-derived work yield.
- Average work extraction: The advantage is largest at high thermal occupation r because the coherence to be unlocked is then greater.
- Repeatability: Work fluctuations can prevent repumping after every individual run, so the protocol instead groups sufficiently many extractions before repumping.
- Repeatability: When failure occurs in this grouped strategy, extra work is invested to ensure perfect repeatability without destroying the reference’s coherence.
B. Single-shot work extraction
The protocol uses a fully quantum, single-shot treatment to extract work from coherence in an individual state without an unbounded reference or asymptotically many runs. Repeatable machines can improve success probability, approaching deterministic extraction with increasingly high-quality references, while finite resources remain below the classical limit.
- Protocol: The protocol performs ε-deterministic work extraction from |γ⟩ for an individual state without an unbounded reference or asymptotically many runs.In the absence of an external coherence source, the state is indistinguishable from γS and the established extraction results apply.
- Coherence advantage: A repeatable thermal machine can reduce single-shot failure probability by exploiting the coherence of |γ⟩.The improvement requires the machine quality ⟨¯∆⟩ to exceed a threshold.
- Coherence advantage: As ⟨¯∆⟩ increases, the failure probability falls from r to r − δε and can reach zero for an unbounded reference.Here, r denotes the thermal occupation of an excited state.
- Resource boundary: The optimal coherence-to-work conversion is accessible only with unbounded coherence resources, although bounded thermal machines can approach it arbitrarily closely.The approach uses a sequence of bounded machines rather than a physically unbounded reference.
- Resource boundary: Dropping classicality assumptions yields improvements over incoherent protocols, but these protocols do not attain classical-limit performance.The analysis emphasizes accounting for the resources composing the thermal machine and preserving them under repeatability.
Appendix A - Collective processing regime
Collective processing or quantum memory can unlock work from coherence that is locked under individual processing. In the asymptotic collective regime, deterministic work per copy reaches the free-energy difference, while a black-box individual interface can conceal collective processing internally.
- Collective processing regime: Collective processing extracts work from relational degrees of freedom in decoherence-free subspaces.One coherent copy can serve as a reference for another, so D(ρS⊗2) differs from D(ρS)⊗2.
- Collective processing regime: For finitely many copies, a non-zero amount of work is unlocked from coherence.The asymptotic result concerns deterministic extraction per copy from infinitely many i.i.d. copies.
- Collective processing regime: In the infinite-copy limit, deterministic work per copy equals F(ρS) − F(γS).This is the collective-processing free-energy limit.
- Black-box implementation: A black-box device can accept individual states and return thermalized states while producing average work F(ρS) − F(γS).Its apparent individual processing can instead rely on a large quantum memory and later collective restoration.
- Black-box implementation: Although externally individual, the black-box protocol hides collective relational processing inside its quantum memory.The outside behavior therefore does not reveal how coherence is processed internally.
Appendix C - Details of the repumping stage
The repumping stage restores the reference using a joint energy-conserving unitary with a weight system. This accounts for repeatability by investing one unit of work when the reference must be returned to its usable state.
- Repumping implementation: Repumping can be implemented by a joint energy-conserving unitary between a weight in |1⟩ and the reference.The construction uses the Pauli X operator in the unitary protocol.
- Repumping implementation: After repumping, the weight ends in |0⟩ and the reference is restored to the state specified by Eq. (10).The reference initially has no population in the ground state.
- Repeated use: The reference’s total free-energy change over repeated protocol uses determines its average free-energy change in the limit M →∞.The reference is compared with its thermal state γR.
Appendix E - Details of approaching free energy limit
The repumping analysis quantifies how a bounded reference can support repeated extraction while controlling the probability of failure. Measurement and memory-erasure costs can be made negligible per input copy as the number of repetitions grows.
- Reference evolution: The reference state evolves through a recurrence while preserving the machine quality ⟨¯∆⟩ throughout the protocol.Initially having no support on the first M energy levels permits extraction from M qubits before ground-state overlap occurs.
- Repumping control: The selective measurement can be implemented energy-conservingly with an ancillary memory, but erasing that memory carries the thermodynamic cost.The measurement distinguishes the low-energy projector PM from its complement.
- Repumping control: The erasure cost can become arbitrarily small as psucc → 1, and its cost per copy scales as M^-1.The cost is paid only after extracting work from M copies.