Fine-Grained Uncertainty Relations for Quantum Testers

TL;DR

This paper extends fine-grained uncertainty relations to quantum testers, providing a framework that includes state measurements and quantum processes, with explicit bounds and applications involving entangled states.

Contribution

It develops FGURs for quantum testers, generalizing the uncertainty principle to quantum processes and offering explicit bounds and specific case analyses.

Findings

Derived FGURs for quantum testers involving entangled states

Provided estimates for the bounds of generalized FGURs

Explicit forms of FGURs for specific quantum testing scenarios

Abstract

The uncertainty principle is one of the features of quantum theory. Fine-grained uncertainty relations (FGURs) are a contemporary interpretation of this principle. Each FGUR is derived from a scenario where multiple measurements of a quantum state are stochastically performed. While state measurements are fundamental, measuring quantum processes, namely, completely positive and trace preserving maps, is also crucial both theoretically and practically. These measurements are mathematically characterized by quantum testers. In this study, we develop FGURs in terms of quantum testers. Because state preparation is a type of quantum process, our framework encompasses the conventional case as a special instance. The generalized FGURs' bounds are typically challenging to compute. Thus, we also provide estimates for these bounds. Specifically, we explore quantum testers involving maximally entangled states in detail. Consequently, some FGURs for quantum testers are derived as explicit forms for specific settings.