Wigner-Araki-Yanase Theorem for Continuous and Unbounded Conserved Observables

Area of Science:

• Quantum Mechanics
• Mathematical Physics

Background:

• The Wigner-Araki-Yanase (WAY) theorem connects conservation laws with measurement commutativity.
• Existing proofs are limited to bounded or discrete observables, excluding continuous ones like momentum.

Purpose of the Study:

• To generalize the WAY theorem for unbounded and continuous conserved observables.
• To investigate the implications for exact quantum measurements and unitary channel implementations.

Main Methods:

• Extension of the WAY theorem under the Yanase condition for unbounded observables.
• Analysis of projective measurements and unitary channel implementations in quantum systems.

Main Results:

• The generalized WAY theorem is established for continuous observables.
• Exact projective position measurements under momentum conservation are shown to be impossible.
• Unitary channel implementations under conservation laws depend on the spectrum of the conserved observable.

Conclusions:

• The study provides a more comprehensive understanding of the WAY theorem's applicability.
• It demonstrates fundamental limitations in performing certain quantum measurements exactly.
• Findings have implications for quantum information processing and experimental quantum mechanics.