Classicality condition on a system observable in a quantum measurement and a relative-entropy conservation law

Abstract

We consider the information flow on a system's observable $X$ corresponding to a positive-operator valued measure under a quantum measurement process $Y$ described by a completely positive instrument from the viewpoint of the relative entropy. We establish a sufficient condition for the relative-entropy conservation law which states that the averaged decrease in the relative entropy of the system's observable $X$ equals the relative entropy of the measurement outcome of $Y$, i.e. the information gain due to measurement. This sufficient condition is interpreted as an assumption of classicality in the sense that there exists a sufficient statistic in a joint successive measurement of $Y$ followed by $X$ such that the probability distribution of the statistic coincides with that of a single measurement of $X$ for the pre-measurement state. We show that in the case when $X$ is a discrete projection-valued measure and $Y$ is discrete, the classicality condition is equivalent to the relative-entropy conservation for arbitrary states. The general theory on the relative-entropy conservation is applied to typical quantum measurement models, namely quantum non-demolition measurement, destructive sharp measurements on two-level systems, a photon counting, a quantum counting, homodyne and heterodyne measurements. These examples except for the non-demolition and photon-counting measurements do not satisfy the known Shannon-entropy conservation law proposed by Ban~(M. Ban, J. Phys. A: Math. Gen. \textbf{32}, 1643 (1999)), implying that our approach based on the relative entropy is applicable to a wider class of quantum measurements.