Quantitative Wigner-Araki-Yanase Theorems for Unitary and Antiunitary Symmetries

Abstract

Symmetry imposes fundamental constraints on quantum measurement and control. The Wigner-Araki-Yanase theorem and its quantitative extensions capture this restriction for continuous symmetries, in terms of fluctuations of conserved generators. Such generator-based bounds, however, do not provide quantitative limitations for discrete unitary symmetries or for antiunitary symmetries. Here we establish quantitative WAY-type theorems for symmetry-breaking projective measurements and unitary gates under arbitrary unitary and antiunitary symmetries. Our approach is based on a two-target no-programming inequality: if a single processor approximately implements two operations that amplify distinguishability, then the corresponding program states must themselves be distinguishable. Applied to symmetric implementations, this converts the error of an asymmetric measurement or gate directly into a lower bound on the asymmetry of the apparatus state, quantified by its fidelity with its symmetry-transformed copy. Our results apply to discrete and antiunitary symmetries, thereby providing a fundamental limit for symmetry-limited quantum measurement and control beyond the continuous-symmetry regime.