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arXiv:2402.16964 · Physical Review A

Zero-fluctuation quantum work extraction

Raffaele Salvia, Vittorio Giovannetti

Published Jul 12, 2024• 3 citations
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Abstract

We study the possibility of deterministic protocols for extracting work from quantum systems. Focusing on the two-point measurement work extraction scenario, we prove that, with enough copies of the system, such zero-fluctuation protocols always exist if the Hamiltonian has a rational spectrum. Leveraging this result, we show that for any Hamiltonian, it is possible to construct a unitary driving protocol on sufficiently many copies of the system with work fluctuations strictly bounded within an arbitrary interval $\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}$, albeit requiring exponentially many copies in $1/\ensuremath{\delta}$.

DOI
10.1103/physreva.110.012213

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We study the possibility of deterministic protocols for extracting work from quantum systems. Focusing on the two-point measurement work extraction scenario, we prove that, with enough copies of the system, such zero-fluctuation protocols always exist if the Hamiltonian has a rational spectrum. Leveraging this result, we show that for any Hamiltonian, it is possible to construct a unitary driving protocol on sufficiently many copies of the system with work fluctuations strictly bounded within an arbitrary interval \ifmmode±\else\textpm\fi\ensuremathδ\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta}\ifmmode±\else\textpm\fi\ensuremathδ, albeit requiring exponentially many copies in 1/\ensuremathδ1/\ensuremath{\delta}1/\ensuremathδ.
There has been an increasing interest in the quantification of nearly deterministic work extraction from a finite number of copies of microscopic particles in finite time. This paradigm, the so-called single-shot $\ensuremath{\epsilon}$-deterministic work extraction, considers processes with small failure probabilities. However, the resulting fluctuations in the extracted work entailed by this failure probability have yet to be studied. In the standard thermodynamics, paradigm fluctuation theorems are powerful tools to study fluctuating quantities. Given that standard fluctuation theorems are inadequate for a single-shot scenario, here we formulate and prove a fluctuation relation specific to the single-shot $\ensuremath{\epsilon}$-deterministic work extraction to bridge this gap. Our results are general in the sense that we allow the system to be in contact with the heat bath at all times. As a corollary of our theorem, we derive the known bounds on the $\ensuremath{\epsilon}$-deterministic work.
Abstract We consider work extraction from N copies of a quantum system. When the same work-extraction process is implemented on each copy, the relative size of fluctuations is expected to decay as <?CDATA 1/N1/\sqrt{N}1/N​?> 1 / N . Here, we consider protocols where the copies can be processed collectively, and show that in this case work fluctuations can disappear exponentially fast in N . As a consequence, a considerable proportion of the average extractable work <?CDATA W{ \mathcal W }W?>  can be obtained almost deterministically by globally processing a few copies of the state. This is derived in the two canonical scenarios for work extraction: (i) in thermally isolated systems, where <?CDATA W{ \mathcal W }W?>  corresponds to the energy difference between initial and passive states, known as the ergotropy, and (ii) in the presence of a thermal bath, where <?CDATA W{ \mathcal W }W?>  is given by the free energy difference between initial and thermal states.
We consider the task of extracting work from quantum systems in the resource theory perspective of thermodynamics, where free states are arbitrary thermal states, and allowed operations are energy conserving unitary transformations. Taking as our work storage system a 'weight' we prove the second law and then present simple protocols which extract average work equal to the free energy change of the system - the same amount as in classical thermodynamics. Crucially, for systems in 'classical' states (mixtures of energy eigenstates) our protocol works on a single copy of the system. This is in sharp contrast to previous results, which showed that in case of almost-deterministic work extraction, collective actions on multiple copies are necessary to extract the free energy. This establishes the fact that free energy is a meaningful notion even for individual systems in classical states. However, for non-classical states, where coherences between energy levels exist, we prove that collective actions are necessary, so long as no external sources of coherence are used.
Abstract We consider work extraction from N copies of a quantum system. When the same work-extraction process is implemented on each copy, the relative size of fluctuations is expected to decay as 1 / N . Here, we consider protocols where the copies can be processed collectively, and show that in this case work fluctuations can disappear exponentially fast in N. As a consequence, a considerable proportion of the average extractable work  can be obtained almost deterministically by globally processing a few copies of the state. This is derived in the two canonical scenarios for work extraction: (i) in thermally isolated systems, where  corresponds to the energy difference between initial and passive states, known as the ergotropy, and (ii) in the presence of a thermal bath, where  is given by the free energy difference between initial and thermal states.
Recent developments of the analysis of the second law of thermodynamics by using the information theoretic quantities has led to the discovery of the so-called nonequilibrium free energy and has allowed us to formulate thermodynamic relations for arbitrary nonequilibrium initial and final states.By using the nonequilibrium free energies, we can quantify thermodynamic costs of information encoding and erasure processes as well as to quantify the extractable work from information heat engines in a unified way.In those general situations, reducing dissipation allows us to increase the extractable work from the system (or to reduce the work cost), and reducing work fluctuation allows us to narrow the work probability distribution and to extract a fluctuation-free work from the system.Suppressing both work fluctuation and dissipation as much as possible is vital to control mesoscopic and nano systems that work at the level of thermal fluctuations.Previous studies have searched for a vanishing work fluctuation protocol by using techniques of quantum information theory, known as the single-shot statistical mechanics (and sometimes called the deterministic work extraction protocol) and the vanishing dissipation protocol, known as the thermodynamically reversible protocol by using the second law of thermodynamics.In this thesis, we address issues concerning a method to control nonequilibrium systems by using fluctuation theorems and trade-off relation between work fluctuation and fluctuation in dissipation.We derive quantum fluctuation theorems under measurement and feedback control, which allow us to study how to suppress dissipation during a general quantum measurement and a feedback control.We then consider a method to suppress work fluctuation and fluctuation in dissipation simultaneously as much as possible by showing the trade-off relation between work and dissipation in terms of their standard deviations and derive an explicit protocol that achieves the lower bound of the trade-off relation.We show that the explicit protocol reproduces the deterministic work extraction protocol in the limit of vanishing work fluctuation and the thermodynamically reversible protocol in the limit of vanishing fluctuation in dissipation.
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.