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

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