Quantum Limits of Measurements Induced by Multiplicative Conservation Laws: Extension of the Wigner-Araki-Yanase Theorem

TL;DR

This paper extends the Wigner-Araki-Yanase theorem to multiplicative conservation laws, revealing fundamental measurement limitations and quantifying the noise bounds associated with multiplicative conserved quantities in quantum systems.

Contribution

It generalizes the WAY theorem to multiplicative conservation laws and derives bounds on measurement noise, highlighting differences from additive conservation laws.

Findings

Measurement of observables not commuting with the modulus of a multiplicatively conserved quantity is limited.

A lower bound for measurement noise in the presence of multiplicative conservation laws is established.

Overcoming measurement noise requires increasing the relative fluctuation of the conserved quantity.

Abstract

The Wigner-Araki-Yanase (WAY) theorem shows that additive conservation laws limit the accuracy of measurements. Recently, various quantitative expressions have been found for quantum limits on measurements induced by additive conservation laws, and have been applied to the study of fundamental limits on quantum information processing. Here, we investigate generalizations of the WAY theorem to multiplicative conservation laws. The WAY theorem is extended to show that an observable not commuting with the modulus of, or equivalently the square of, a multiplicatively conserved quantity cannot be precisely measured. We also obtain a lower bound for the mean-square noise of a measurement in the presence of a multiplicatively conserved quantity. To overcome this noise it is necessary to make large the coefficient of variation (the so-called relative fluctuation), instead of the variance as is the case for additive conservation laws, of the conserved quantity in the apparatus.