Quantifying unsharpness of observables in an outcome-independent way

Abstract

Recently, a very beautiful measure of the unsharpness (fuzziness) of the observables is discussed in the paper [Phys. Rev. A 104, 052227 (2021)]. The measure which is defined in this paper is constructed via uncertainty and does not depend on the values of the outcomes. There exist several properties of a set of observables (e.g., incompatibility, non-disturbance) that do not depend on the values of the outcomes. Therefore, the approach in the above-said paper is consistent with the above-mentioned fact and is able to measure the intrinsic unsharpness of the observables. In this work, we also quantify the unsharpness of observables in an outcome-independent way. But our approach is different than the approach of the above-said paper. In this work, at first, we construct two Luder's instrument-based unsharpness measures and provide the tight upper bounds of those measures. Then we prove the monotonicity of the above-said measures under a class of fuzzifying processes (processes that make the observables more fuzzy). This is consistent with the resourcetheoretic framework. Then we relate our approach to the approach of the above-said paper. Next, we try to construct two instrument-independent unsharpness measures. In particular, we define two instrument-independent unsharpness measures and provide the tight upper bounds of those measures and then we derive the condition for the monotonicity of those measures under a class of fuzzifying processes and prove the monotonicity for dichotomic qubit observables. Then we show that for an unknown measurement, the values of all of these measures can be determined experimentally. Finally, we present the idea of the resource theory of the sharpness of the observables.