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Fluctuations in measurement-based quantum engines and quantum batteries derived from the Kirkwood-Dirac quasiprobability distribution

Abdelkader El Makouri, Abdallah Slaoui, Rachid Ahl Laamara

Published Apr 30, 2026• 0 citations

Abstract

In this paper, we consider fluctuations in a three-stroke cycle based on quantum measurement. In the first stroke, the engine is fueled by a nonselective quantum measurement; in the second stroke, an appropriate unitary operation is applied to extract work; and finally, the working medium is thermalized to complete the cycle. This protocol is analogous to charging, then discharging, and finally thermalizing a quantum battery. Using the Kirkwood-Dirac quasiprobability, we derive equations that provide insights into the fluctuations of heat and work, and thus into the charging and discharging processes. We then propose a quantum circuit to measure the cumulants of work and heat using a qubit ancilla. Afterward, we derive important upper and lower bounds on fluctuations in heat; this is thanks to the way we decompose heat fluctuations and the Kadison-Schwarz inequality. We also show, for the first time, that the imaginary part of the second cumulant of work equals the linear response of an average work (to be defined in the main text) or the quantum cross entropy. We further introduce an operator whose average defines the battery capacity and whose variance quantifies the corresponding fluctuations of the battery capacity. Our analytical results are illustrated and discussed through representative examples. We believe that our study will, on the one hand, be useful for understanding fluctuations in energy extracted from and injected into quantum batteries based on quantum measurements and, on the other hand, contribute to the foundations of quantum thermodynamics. More generally, our results can be applied beyond the three-stroke cycle considered here, especially to quantum batteries subject to environmental noise and decoherence.

DOI: 10.1103/fls6-mxgq

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In this paper, we consider fluctuations in a three-stroke cycle based on quantum measurement. In the first stroke, the engine is fueled by a nonselective quantum measurement; in the second stroke, an appropriate unitary operation is applied to extract work; and finally, the working medium is thermalized to complete the cycle. This protocol is analogous to charging, then discharging, and finally thermalizing a quantum battery. Using the Kirkwood-Dirac quasiprobability, we derive equations that provide insights into the fluctuations of heat and work, and thus into the charging and discharging processes. We then propose a quantum circuit to measure the cumulants of work and heat using a qubit ancilla. Afterward, we derive important upper and lower bounds on fluctuations in heat; this is thanks to the way we decompose heat fluctuations and the Kadison-Schwarz inequality. We also show, for the first time, that the imaginary part of the second cumulant of work equals the linear response of an average work (to be defined in the main text) or the quantum cross entropy. We further introduce an operator whose average defines the battery capacity and whose variance quantifies the corresponding fluctuations of the battery capacity. Our analytical results are illustrated and discussed through representative examples. We believe that our study will, on the one hand, be useful for understanding fluctuations in energy extracted from and injected into quantum batteries based on quantum measurements and, on the other hand, contribute to the foundations of quantum thermodynamics. More generally, our results can be applied beyond the three-stroke cycle considered here, especially to quantum batteries subject to environmental noise and decoherence.

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The charging of a quantum battery by a four-stroke quantum machine that works either as an engine or a refrigerator is investigated. The presented analysis provides the energetic behavior of the combined system in terms of the heat- and workflows of the machine, the average, the variance of the battery's energy, as well as the coherent and incoherent parts of its ergotropy. To monitor the battery state, its energy is measured either after the completion of any cycle or after a prescribed number of cycles is carried out. The resulting battery performances greatly differ for those two cases. During the first charging epoch with an engine, the regular measurements speed up the charging, whereas the gain of ergotropy is more pronounced in the absence of measurements. In a later stage, the engine fails to work as such, while it still continues charging the battery that eventually reaches the maximally charged state in the absence of intermediate measurements and a suboptimally charged state for a periodically measured battery. For a refrigerator, the charging of the measured battery also proceeds faster during the first epoch. Only during the second stage, when the machine fails to extract heat from the cold bath, the influence of the measurements becomes less pronounced, leading to rather similar asymptotic states for the two measurement scenarios.

We study the connection between the charging power of quantum batteries and the fluctuations of the extractable work. We prove that in order to have a nonzero rate of change of the extractable work, the state ${\ensuremath{\rho}}_{\mathcal{W}}$ of the battery cannot be an eigenstate of a ``free energy operator,'' defined by $\mathcal{F}\ensuremath{\equiv}{H}_{\mathcal{W}}+{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{log}({\ensuremath{\rho}}_{\mathcal{W}})$, where

{H}_{\mathcal{W}}

is the Hamiltonian of the battery and $\ensuremath{\beta}$ is the inverse temperature of a reference thermal bath with respect to which the extractable work is calculated. We do so by proving that fluctuations in the free energy operator upper bound the charging power of a quantum battery. Our findings also suggest that quantum coherence in the battery enhances the charging process, which we illustrate on a toy model of a heat engine.

Abstract In this work, we introduce a three-stroke quantum engine with a single-qubit working substance whose cycle consists of two strokes arise due to performing two distinct general quantum measurements and it is completed by thermalization through contact with a finite temperature thermal reservoir. It is demonstrated that energy is imported into the engine by first measurement channel and work (useful energy) is extracted from it, in a frictionless isentropic manner, by the second measurement channel. It is particularly shown that the engine is capable to have perfect efficiency. In continuation, we equip the engine cycle with two additional adiabatic strokes. As illustrated, the presence of adiabatic strokes in the cycle provide an improved lower bound for the parametrized efficiency range of the engine.

One of the most fundamental tasks in quantum thermodynamics is extracting energy from one system and subsequently storing this energy in an appropriate battery. Both of these steps, work extraction and charging, can be viewed as cyclic Hamiltonian processes acting on individual quantum systems. Interestingly, so-called passive states exist, whose energy cannot be lowered by unitary operations, but it is safe to assume that the energy of any not fully charged battery may be increased unitarily. However, unitaries raising the average energy by the same amount may differ in qualities such as their precision, fluctuations, and charging power. Moreover, some unitaries may be extremely difficult to realize in practice. It is hence of crucial importance to understand the qualities that can be expected from practically implementable transformations. Here, we consider the limitations on charging batteries when restricting to the feasibly realizable family of Gaussian unitaries. We derive optimal protocols for general unitary operations as well as for the restriction to easier implementable Gaussian unitaries. We find that practical Gaussian battery charging, while performing significantly less well than is possible in principle, still offers asymptotically vanishing relative charge variances and fluctuations.

We study the work fluctuations in ergotropic heat engines, namely two-stroke quantum Otto engines where the work stroke is designed to extract the ergotropy (the maximum amount of work by a cyclic unitary evolution) from a couple of quantum systems at canonical equilibrium at two different temperatures, whereas the heat stroke thermalizes back the systems to their respective reservoirs. We provide an exhaustive study for the case of two qutrits whose energy levels are equally spaced at two different frequencies by deriving the complete work statistics. By varying the values of temperatures and frequencies, only three kinds of optimal unitary strokes are found: the swap operator U1, an idle swap U2 (where one of the qutrits is regarded as an effective qubit), and a non-trivial permutation of energy eigenstates U3, which indeed corresponds to the composition of the two previous unitaries, namely U3=U2U1. While U1 and U2 are Hermitian (and hence involutions), U3 is not. This point has an impact on the thermodynamic uncertainty relations (TURs), which bound the signal-to-noise ratio of the extracted work in terms of the entropy production. In fact, we show that all TURs derived from a strong detailed fluctuation theorem are violated by the transformation U3.