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Energy extraction from quantum batteries by means of completely positivetrace-preserving (CPTP) maps is quite well-studied in the literature. Itnaturally leads to the concept of CPTP-local passive states, which identifybipartite states from which no energy can be squeezed out by applying any CPTPmap to a particular subsystem. We prove, for arbitrary dimension, that if astate is CPTP-local passive with respect to a Hamiltonian, then an arbitrarynumber of copies of the same state - including an asymptotically large one - isalso CPTP-local passive. We show further that energy can be extractedefficiently from these CPTP-local passive states employing non-completelypositive trace-preserving (NCPTP) but still physically realizable maps on thesame part of the shared battery on which operation of CPTP maps were useless.Moreover, we provide the maximum extractable energy using local-CPTPoperations, and then, we present an explicit class of states and correspondingHamiltonians, for which the maximum can be outperformed using physical localNCPTP maps. We build an analogy between the relative status of CPTP and NCPTPoperations for energy extraction in quantum batteries, and the association ofdistillable entanglement with entanglement cost for asymptotic localmanipulations of entanglement. The surpassing of the maximum energy extractableby NCPTP maps for CPTP passive as well as for CPTP non-passive battery statescan act as detectors of non-CPTPness of quantum maps. Finally, we provide anecessary condition for an arbitrary bipartite state to be unable to supply anyenergy using NCPTP operations on one party with respect to an arbitrary butfixed Hamiltonian.

Non-completely positive quantum maps enable efficient local energy extraction in batteries