Positive and non-positive measurements in energy extraction from quantum batteries

We extend the concept of stochastic energy extraction from quantum batteries to the scenario where both positive operator-valued (POV) and physically realizable non-positive operator-valued measurements (NPOVMs) are applied on the auxiliary connected to the battery in presence of noise. The process involves joint evolution of the battery and the auxiliary, an interaction of the auxiliary with its environment which induces noise in the auxiliary, and performing a POVM or NPOVM on the auxiliary, and finally the selection of a particular measurement outcome. Application of POVM on the auxiliary can be realized by attaching an external system to the auxiliary, which is initially in a product state with the rest of the system, and performing a joint projective measurement on the auxiliary and external. On the other hand, NPOVMs can be performed in two ways: if there are interactions leading to correlations among the auxiliary and environment, then performing a projective measurement on the auxiliary environment system can be interpreted as a physically realizable NPOVM on the auxiliary. Moreover, if there exists interaction among auxiliary, environment and external initially then the global measurement on the auxiliary, environment and external is also effectively an NPOVM on the auxiliary. We find the expressions of the maximum stochastically extractable energy by performing POVMs and NPOVMs of both types on the auxiliary and show that the latter does not depend on the applied noise. Focusing on a particular model of a battery, auxiliary, environment, and external, all being qubits, we show that stochastically extractable energy by POVMs is less than or equal to the same by type-1 NPOVM under various noise models. We additionally consider the case when a limited set of measurement operators is allowed and compare the accessible energy using these restricted set of POVM and NPOVM of type-1.