  Energy extraction from quantum batteries by means of completely positive
trace-preserving (CPTP) maps leads to the concept of CPTP-local passive states,
which identify bipartite states from which no energy can be squeezed out by
applying any CPTP map to a particular subsystem. We prove, for arbitrary
dimension, that if a state is CPTP-local passive with respect to a Hamiltonian,
then an arbitrary number of copies of the same state - including an
asymptotically large one - is also CPTP-local passive. We show further that
energy can be extracted efficiently from CPTP-local passive states employing
NCPTP but still physically realizable maps on the same part of the shared
battery on which operation of CPTP maps were useless. Moreover, we provide the
maximum extractable energy using local-CPTP operations, and then, we present an
explicit class of states and corresponding Hamiltonians, for which the maximum
can be outperformed using physical local NCPTP maps. We provide a necessary and
sufficient condition and a separate necessary condition for an arbitrary
bipartite state to be unable to supply any energy using non-completely positive
trace-preserving (NCPTP) operations on one party with respect to an arbitrary
but fixed Hamiltonian. We build an analogy between the relative status of CPTP
and NCPTP operations for energy extraction in quantum batteries, and the
association of distillable entanglement with entanglement cost for asymptotic
local manipulations of entanglement. The surpassing of the maximum energy
extractable by NCPTP maps for CPTP-passive as well as for CPTP non-passive
battery states can act as detectors of non-CPTPness of quantum maps.
