Uncertainty Relations for coarse-grained measurements: an overview

TL;DR

This paper reviews the development and application of uncertainty relations tailored for coarse-grained measurements in continuous variable quantum systems, emphasizing both theoretical foundations and experimental perspectives.

Contribution

It provides a comprehensive overview of the current state of coarse-grained uncertainty relations and their relevance to quantum physics and information tasks.

Findings

Development of uncertainty relations for coarse-grained observables

Applications to quantum correlations and cryptography

Balance of theoretical and experimental insights

Abstract

Uncertainty relations involving complementary observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continnuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, a number of uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90$^{th}$ anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.