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

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