A geometrical framework for quantum incompatibility resources

Abstract

Quantum incompatibility is a fundamental property in quantum physics and is considered as a resource in quantum information processing tasks. Here, we construct a framework based on the incompatibility witness originated from quantum state discrimination. In this framework, we discuss the geometrical properties of the witnesses with noisy mutually unbiased bases and construct a quantifier of quantum incompatibility associated with geometrical information. Furthermore, we explore the incompatibility of a pair of positive operator valued measurements, which only depends on the information of measurements, and discuss the incompatibility of measurements which can be discriminated by mutually unbiased bases. Finally, if we take a resource-theory perspective, the new-defined quantifier can characterize the resource of incompatibility. This geometrical framework gives the evidence that vectors can be utilized to describe resourceful incompatibility and make a step to explore geometrical features of quantum resources.