Auxiliary-assisted energy distillation from quantum batteries

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.