  Completely positive and trace-preserving maps characterize physically
implementable quantum operations. On the other hand, general quantum maps, such
as positive but not completely positive maps, which can not be physically
implemented, are fundamental ingredients in quantum information, both in
theoretical and practical perspectives. It motivates how well one can
approximate a general quantum map by physically implementable operations. In
this work, we introduce a systematic framework to resolve this task using the
quasiprobability decomposition technique. More specifically, we decompose a
target quantum map into a linear combination of physically implementable
operations and introduce the physical implementability measure as the least
amount of negative portion that the quasiprobability must pertain. We show this
measure is efficiently computable by semidefinite programs and prove several
properties of this measure, such as faithfulness, additivity, and unitary
invariance. We derive lower and upper bounds in terms of the Choi operator's
trace norm and obtain analytic expressions for several quantum maps of
practical interests. Furthermore, we endow this measure an operational meaning
within the quantum error mitigation scenario, quantifying the ultimate sampling
cost achievable using the full expressibility of quantum computers. In
particular, for parallel quantum noises, we show that global error mitigation
has no advantage over local error mitigation.
