In the conventional axiomatic formalism of quantum mechanics, a physical measurement of a quantum observable is mathematically represented by the spectral decomposition of the Hermitian operator associated with the observable. This constitutes the scenario of a Lüders (projective) measurement, which includes a von Neumann (rank-one projective) measurement as a special and prominent instance. In this context, the measurement is called sharp in the sense that each measurement operator is an orthogonal projection (eigenprojection of the observable). In the modern operational formalism, a measurement is represented by a positive-operator valued measure (POVM), which consists of a family of non-negative definite operators (measurement operators, effects) summing to the identity. In this scenario, a measurement is called unsharp (fuzzy) if some measurement operators are not orthogonal projections. A natural question arises as to how to quantify unsharpness of a measurement. In this work, we address this issue in terms of uncertainty. For this purpose, we study a family of observables associated with a measurement and their uncertainty. By exploiting the difference between the (total) measurement uncertainty and the observable uncertainty, we are led to some information-theoretic quantifiers of unsharpness. We reveal their basic properties and illustrate them through some important measurements. In particular, we characterize Lüders measurements and equiangular POVMs as extreme measurements in terms of unsharpness.

• Received 14 October 2021
• Revised 4 November 2021
• Accepted 16 November 2021

DOI:https://doi.org/10.1103/PhysRevA.104.052227