Coherence-variance uncertainty relation and coherence cost for quantum measurement under conservation laws

TL;DR

This paper enhances the uncertainty relation in quantum mechanics and quantifies the coherence required for quantum measurements under conservation laws, improving existing limitations and providing precise coherence costs.

Contribution

It systematically refines the Kennard-Robertson uncertainty relation and determines the exact coherence needed for quantum measurements under conservation laws.

Findings

Improved the Kennard-Robertson uncertainty relation.

Quantified the coherence cost for quantum measurements under conservation laws.

Provided an asymptotic equality for coherence requirements.

Abstract

Uncertainty relations are one of the fundamental principles in physics. It began as a fundamental limitation in quantum mechanics, and today the word {\it uncertainty relation} is a generic term for various trade-off relations in nature. In this letter, we improve the Kennard-Robertson uncertainty relation, and clarify how much coherence we need to implement quantum measurement under conservation laws. Our approach systematically improves and reproduces the previous various refinements of the Kennard-Robertson inequality. As a direct consequence of our inequalities, we improve a well-known limitation of quantum measurements, the Wigner-Araki-Yanase-Ozawa theorem. This improvement gives an asymptotic equality for the necessary and sufficient amount of coherence to implement a quantum measurement with the desired accuracy under conservation laws.