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arXiv:2307.16856 · Physical Review A

Auxiliary-assisted energy distillation from quantum batteries

Paranjoy Chaki, Aparajita Bhattacharyya, Kornikar Sen, Ujjwal Sen

Published Nov 21, 2025• 6 citations
arXivHTMLPDF

Abstract

We discuss the idea of extracting energy from a quantum battery, applying a projective measurement on an auxiliary system. The battery is initially connected to the auxiliary system and allowed to interact with it. After some time, we execute a measurement on the auxiliary system which probabilistically projects the setup to a particular state, and the corresponding state of the battery is the final state. We consider the sum of the product of the energy difference between the initial and final states of the battery with the probability of getting that final state, where the sum is taken over all the preferable outcomes, that is, the outcomes which reduce the energy of the battery. We define the maximum value of this quantity as the distillable energy, where the maximization is taken over the time of interaction and auxiliary state and measurement basis parameters. Restricting ourselves to a particular uncountable set of states, we find that distillable energy is always higher than the ergotropy of the battery, irrespective of the presence or absence of entanglement between battery and auxiliary. We also compare the distillable energy with the energy extracted using the interaction between the battery and the auxiliary without any measurements. In comparison with the measurement-free scenario, we show that while measurement-based protocols do not provide any enhancement in the amount of extractable energy, they do yield a distinct advantage in terms of power, most notably in the case of distillable power, surpassing the power obtained without measurements. We show that the average of extracted energy overall measurement outcomes is independent of the applied measurement and equivalent to applying no measurement. Additionally, we find that the advantage in power for measurement-based energy extraction reduces with the increase in the size of the battery if the dimension of the auxiliary is kept fixed. Finally, we find that the ground state is the only state for which distillable energy is zero. We also analyze the behavior of the maximum value (maximization done over the same parameters as in the case of distillable energy) of the product of the probability of getting a particular suitable outcome and the energy difference of the corresponding initial and final states of the battery, which is referred to as maximum probabilistically extractable energy.

DOI
10.1103/cyrc-ms34

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We discuss the idea of extracting energy from a quantum battery, applying a projective measurement on an auxiliary system. The battery is initially connected to the auxiliary system and allowed to interact with it. After some time, we execute a measurement on the auxiliary system which probabilistically projects the setup to a particular state, and the corresponding state of the battery is the final state. We consider the sum of the product of the energy difference between the initial and final states of the battery with the probability of getting that final state, where the sum is taken over all the preferable outcomes, that is, the outcomes which reduce the energy of the battery. We define the maximum value of this quantity as the distillable energy, where the maximization is taken over the time of interaction and auxiliary state and measurement basis parameters. Restricting ourselves to a particular uncountable set of states, we find that distillable energy is always higher than the ergotropy of the battery, irrespective of the presence or absence of entanglement between battery and auxiliary. We also compare the distillable energy with the energy extracted using the interaction between the battery and the auxiliary without any measurements. In comparison with the measurement-free scenario, we show that while measurement-based protocols do not provide any enhancement in the amount of extractable energy, they do yield a distinct advantage in terms of power, most notably in the case of distillable power, surpassing the power obtained without measurements. We show that the average of extracted energy overall measurement outcomes is independent of the applied measurement and equivalent to applying no measurement. Additionally, we find that the advantage in power for measurement-based energy extraction reduces with the increase in the size of the battery if the dimension of the auxiliary is kept fixed. Finally, we find that the ground state is the only state for which distillable energy is zero. We also analyze the behavior of the maximum value (maximization done over the same parameters as in the case of distillable energy) of the product of the probability of getting a particular suitable outcome and the energy difference of the corresponding initial and final states of the battery, which is referred to as maximum probabilistically extractable energy.
Abstract The reduced state of a small system strongly coupled to a charger in thermal equilibrium may be athermal and used as a small battery once disconnected. By harnessing the battery-charger correlations, the battery’s extractable energy can increase above the ergotropy. We introduce a protocol that uses a quantum system as a memory that measures the charger and leaves the battery intact in its charged state. Using the information gained from the measurement, the daemonic ergotropy of the battery is extracted. Then the battery is reconnected to the charger, thermalizing and charging it. However, the memory should return to its initial standard state to close the thermodynamic cycle. Thus, on the one hand, the work cost of the cycle is the sum of the disconnecting and reconnecting battery-charger work plus the measurement and erasure work. On the other hand, the extracted energy is the daemonic ergotropy of the battery plus the ergotropy of the memory. The ratio of these quantities defines the efficiency of the cycle. The protocol is exemplified by a modified transverse spin 1/2 Ising chain, one spin functioning as the battery and the others as the charger. The memory is another auxiliary spin 1/2. We found pairs of measurement schemes from which we extract the same daemonic ergotropy from the battery, they dissipate the same amount of energy, and one leaves the memory in an active state, the other in a passive state. We study the memory’s ergotropy and the daemonic ergotropy of the battery. We find that with measurements, the efficiency can surpass that of the unmeasured protocol, given conditions on temperature, coupling, and choice of the measurement operators.
Postselection can enhance the performance of a selected quantum battery branch, raising the question of whether the optimum can always be achieved by product measurements on the auxiliary subsystems. For battery auxiliary states obtained by mixing a pure state with global white noise of strength p, the conditional ergotropy reduces to a generalized Rayleigh quotient. For 0<p<1, product postselection attains the joint optimum if and only if the eigenspace associated with the largest generalized eigenvalue contains a product vector. For two auxiliary qubits, a strict product-joint gap therefore occurs precisely when this eigenspace is one dimensional and spanned by an entangled vector. Noisy W and GHZ states display qualitatively different behavior: the three qubit W state exhibits a strict gap throughout 0<p<1, whereas the three qubit GHZ state admits a product optimum. For interacting two qubit batteries, the optimal postselection can switch from a Bell outcome to a product outcome as the coupling strength is varied. By contrast, the outcome averaged capacity and ergotropy are independent of the measurement basis, showing that the joint measurement advantage is genuinely postselective.
We investigate energy distillation from quantum batteries within the framework of generalized quantum measurements, including both positive operator-valued measurements (POVMs) and physically realizable non-positive operator-valued measurements (NPOVMs) performed on an auxiliary system coupled to the battery. Two classes of NPOVMs, namely type-1 and type-2, are analyzed in the presence of environmental noise acting on the auxiliary system. We derive general expressions for the distillable energy corresponding to positive and non-positive measurements and show that the distillable energy obtained via NPOVMs remains robust against environmental noise. For a specific model, we demonstrate that the energy extracted using type-1 NPOVMs exceeds or equals that obtained via POVMs under amplitude-damping, dephasing, and bit-flip noise, while type-2 NPOVMs outperform POVMs under amplitude-damping noise. These results establish a clear operational advantage of NPOVMs for energy extraction. We also analyze restricted measurement settings and compare the accessible distillable energy for constrained positive and type-1 non-positive measurements.
Thermally charged quantum batteries provide a stable and easy-to-have energy resource, but remain passive and therefore do not allow useful energy extraction on demand via unitary operations. We introduce a time-dependent stirring protocol that activates a thermally charged quantum battery through a time-dependent coupling to an auxiliary activator, thereby generating correlations that drive the battery into an active state. Accounting explicitly for the energetic cost of the stirring process, the entropic cost of correlation generation and the activation time, we derive bounds on the maximal net extractable energy. Incorporating projective measurement on the activator and exploiting the information gained through the measurement further enhances the extractable energy. As an experimentally relevant example, we analyze a waveguide-QED setup where a harmonic-oscillator battery (waveguide) is stirred and monitored by a two-level system. We characterize the performance of the protocol in terms of net extractable energy (net ergotropy) and power output.
The amount of work that can be extracted from a quantum system can be increased by exploiting the information obtained from a measurement performed on a correlated ancillary system. The concept of daemonic ergotropy has been introduced to properly describe and quantify this work extraction enhancement in the quantum regime. We here explore the application of this idea in the context of continuously monitored open quantum systems, where information is gained by measuring the environment interacting with the energy-storing quantum device. We first show that the corresponding daemonic ergotropy takes values between the ergotropy and the energy of the corresponding unconditional state. The upper bound is achieved by assuming an initial pure state and a perfectly efficient projective measurement on the environment, independently of the kind of measurement performed. On the other hand, if the measurement is inefficient or the initial state is mixed, the daemonic ergotropy is generally dependent on the measurement strategy. This scenario is investigated via a paradigmatic example of an open quantum battery: a two-level atom driven by a classical field and whose spontaneously emitted photons are continuously monitored via either homodyne, heterodyne, or photodetection.
In realistic situations, physical systems can not be completely isolated from its environment. Its inevitable interaction with the environment can influence the working process of the device. In this paper, we consider two-qubit quantum batteries where one qubit of the battery is successively interacting with the spins present in the surrounding environment. We examine the effect of the interaction on the maximum amount of energy that can be extracted from the battery using unitaries. In this context, we use the notion of locally passive states. In particular, we examine the behavior of the amount of extractable work from the noisy battery, initially prepared in a locally passive or ordinary pure state, having a fixed initial entanglement, with the number of interactions the qubit has gone through. We also examine the amount of locally extractable work from the noisy battery. We realize though the amount of extractable energy, be it global or local, as a whole will decrease with the number of spins of environment it interacted with, but if we increase the time interval of the interaction with each spin, after a cut off value of the interval, the small time behavior shows a peculiarity, i.e., the extractable energy within a single interaction starts to increase with time. The cut-off time indicates the Markovian-to-non-Markovian transition of the interaction. We also observe a non-Markovian increase in extractable energy from the Markovian scenario.