Separation of measurement uncertainty into quantum and classical parts based on Kirkwood–Dirac quasiprobability and generalized entropy

Abstract

Abstract Measurement in quantum mechanics is notoriously unpredictable. The uncertainty in quantum measurement can arise from the noncommutativity between the state and the measurement basis which is intrinsically quantum, but it may also be of classical origin due to the agent’s ignorance. It is of fundamental as well as practical importance to cleanly separate the two contributions which can be directly accessed using laboratory operations. Here, we propose two ways of decomposition of the total measurement uncertainty additively into quantum and classical parts. In the two decompositions, the total uncertainty of a measurement described by a positive-operator-valued measure (POVM) over a quantum state is quantified respectively by two generalized nonadditive entropies of the measurement outcomes; the quantum parts are identified, respectively, by the nonreality and the nonclassicality—which captures simultaneously both the nonreality and negativity—of the associated generalized Kirkwood–Dirac quasiprobability relative to the POVM of interest and a projection-valued measure and maximized over all possible choices of the latter; and, the remaining uncertainties are identified as the classical parts. Both decompositions are shown to satisfy a few plausible requirements. The minimum of the total measurement uncertainties in the two decompositions over all POVM measurements are given by the impurity of the quantum state quantified by certain generalized quantum entropies, and are entirely classical. We argue that nonvanishing genuine quantum uncertainty in the two decompositions are sufficient and necessary to prove quantum contextuality via weak measurement with postselection. Finally, we suggest that the genuine quantum uncertainty is a manifestation of a specific measurement disturbance.