Abstract
We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an overall composite measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one’s ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the Lüders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement.
Highlights
Ever since Heisenberg proposed uncertainty principle under consideration of γ-ray microscope in [1], the uncertainty principle has become one of the most central concepts in quantum physics
The point to note in the course of the quantifications is that successive measurement scheme has played major roles in clarifying meaning of error and disturbance, and with increasing experimental ability to control quantum systems these relations were proved [9,10] by applying this scheme
We have explicitly explained general description of successive measurement scheme with respect to two scenarios. We have considered this scheme as a method to implement an overall measurement, and, as a result, observed that unsharpness of the overall observable gives limits on one’s ability to measure it with arbitrarily low uncertainty as formulated in Equation (14)
Summary
Ever since Heisenberg proposed uncertainty principle under consideration of γ-ray microscope in [1], the uncertainty principle has become one of the most central concepts in quantum physics. Inspired by the Heisenberg’s first insight, entropic uncertainty relations for successive projective measurements were considered in an information-theoretic approach [12,21] This approach was developed based on Rényi’s entropies [22] and Tsallis’ entropies [23] for a pair of qubit observables. It was argued that the second scenario can be considered as a general method to measure any pair of jointly measurable quantum observables [24], and further it has special usefulness due to so-called universality of successive measurement [25] In this regard, the range of its applications becomes broader (see the references in [25]).
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