Auxiliary-assisted energy distillation from quantum batteries

TL;DR

This paper introduces an auxiliary-assisted, measurement-based scheme for extracting energy from quantum batteries, defining distillable energy as a post-selection–weighted sum over favorable measurement outcomes and optimizing over interaction time, initial auxiliary states, and measurement bases. It shows that distillable energy generally exceeds ergotropy, independent of whether the battery and auxiliary are initially entangled, while the average energy extracted with measurements does not surpass the unitary benchmark; however, the protocol yields a significant power advantage (distillable power), especially when initial entanglement is present. Entanglement boosts the maximal probabilistic extractable energy, though it does not increase distillable energy itself, and the set of measurement-passive states reduces to the ground state under the considered model. For larger batteries, the energy-distillation advantage wanes, but the measured-based approach can still outperform the no-measurement protocol in terms of power, highlighting a trade-off between scale and advantage in energy distillation.

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.

Figures (7)

• Figure 1: Schematic representation of auxiliary-assisted measurement-based energy extraction procedure. The green battery in the diagram represents a single-qubit quantum battery. After connecting the battery to an auxiliary, the entire battery-auxiliary system is evolved under a unitary evolution. An optimal projective measurement is then carried out on the auxiliary system for which the extractable energy would be maximum. The auxiliary is then traced out. It should be mentioned that in order to increase the amount of extractable energy, we additionally optimize the process over the time instance at which the measurement is performed and the initial state of the auxiliary.
• Figure 1: Fig. \ref{['1a']} (a) shows the probabilistically extractable energy $W^{M,F}_S$ in a two-dimensional projection for a fixed measurement setting and auxiliary system (as described in subsection \ref{['2xb']}). Here, the initial state parameter $k$ is plotted along the vertical axis and the interaction time $t$ through the Hamiltonian $H_{BA}$ along the horizontal axis, assuming the battery and auxiliary start in a product state. Fig. \ref{['1a']} (b) presents the corresponding results for initially entangled states, with the horizontal and vertical axes defined as in Fig. \ref{['1a']} (a). In this case, the probablistically extractable energy is denoted as $W^{M,F}_E$. The color bars in the respective plots indicate the values of $W^{M,F}_S$ and $W^{M,F}_E$ for given $t$ and $k$.
• Figure 2: Dependence of extractable energies on the initial state of the battery. In Fig. \ref{['non-linearity-graphx']} (a), we plot $W^U(\rho_B,H_B)$ (blue line), $W^{MD}_S(\rho_B,H_B)$ (pink dot), and $W^{MD}_E(\rho_B,H_B)$ (dark slate gray) along the vertical axis as a function of $k$ presented on the horizontal axis. Here $W^{MD}_S(\rho_B,H_B)$ and $W^{MD}_E(\rho_B,H_B)$ are the terms that denote the distillable energy when the battery and auxiliary are initially product and initially entangled states. On the other hand, in Fig. \ref{['non-linearity-graphx']} (b), $W^U(\rho_B,H_B)$ (blue line), $W^{M}_S(\rho_B,H_B)$ (green line), and $W^{M}_S(\rho_B,H_B)$ (red line) are plotted along the vertical axis, and the horizontal axis represents the initial battery state's parameter $k$. Here the terms $W^{M}_S(\rho_B,H_B)$ and $W^{M}_S(\rho_B,H_B)$ denote the maximum probabilistically extractable energy when the battery and auxiliary are initially product and initially entangled states.
• Figure 3: $W^{D}_S(\rho_B,H_B)-W^U(\rho_B,H_B)$ vs. $k$, $W^{M}_S(\rho_B,H_B)-W^U(\rho_B,H_B)$ vs. $k$ and $W^{M}_E(\rho_B,H_B)-W^{M}_S(\rho_B,H_B)$ Vs $S(\rho_{BA})$ plots are depicted in Fig. \ref{['f3']} (a), Fig. \ref{['f3']} (b), and Fig. \ref{['f3']} (c). Fig. \ref{['f3']} (a) and Fig. \ref{['f3']} (b) $W^{D}_S(\rho_B,H_B)-W^U(\rho_B,H_B)$ and $W^{M}_S(\rho_B,H_B)-W^U(\rho_B,H_B)$, plotted along the vertical axis, with respect to $k$, presented along the horizontal axis. The vertical axis is in the units of $h$, which has the energy unit, and the horizontal axis is again dimensionless. In Fig. \ref{['f3']} (c), we depict $W^{M}_E(\rho_B,H_B)-W^{M}_S(\rho_B,H_B)$ with respect to the entanglement content of the initial state of the battery and the auxiliary. Here the orange and blue curves represent the parameter regions $k>0$ and $k<0$, respectively. The entanglement is quantified using entanglement entropy. The horizontal axis of the main plot is dimensionless, whereas the vertical axes of both of the plots are in the units of $h$, that is, in energy units. The horizontal axis of the inset, i.e., the entanglement entropy, is plotted in units of ebits.
• Figure 4: In Fig. \ref{['4f']} (a), we plot distilled energy $W^{D,F}_S$ in a 2D projection graph for a fixed measurement setting and fixed auxiliary system (given in the last part of subsection \ref{['2xb']}) with the initial state parameter $k$ along the vertical axis and time $t$ of interaction through Hamiltonian $H_{BA}$ along the horizontal axis when the battery and auxiliary are initially product. On the other hand, Fig. \ref{['4f']} (b) describes the same numerical studies for initially entangled states where the horizontal and vertical axes have the same notation as Fig. \ref{['4f']} (a). The distilled energy for a fixed measurement setting and fixed auxiliary system for initially entangled states is denoted as $W^{D,F}_E$. The values of $W^{D,F}_S$ and $W^{D,F}_E$ for a given $t$ and $k$ values are denoted by color bars of respective plots.