Quantum measurements need not conserve energy: relation to the Wigner-Araki-Yanase theorem

TL;DR

This paper investigates the apparent contradiction between quantum measurement energy non-conservation and the Wigner-Araki-Yanase theorem, revealing that certain assumptions lead to physically unlikely conclusions about the measuring apparatus.

Contribution

It demonstrates that under specific assumptions, the Yanase condition follows from the WAY theorem and that the energy operator of the measuring device must be trivial, challenging previous interpretations.

Findings

The Yanase condition is derived from the original WAY theorem hypotheses.

The energy operator of the measuring apparatus must be proportional to the identity.

The assumptions imply physically unlikely constraints on the measuring device.

Abstract

The paper focuses on the fact that quantum projective measurements do not necessarily conserve energy. On the other hand the Wigner-Araki-Yanase (WAY) theorem states that assuming a "standard" von Neumann measurement model and "additivity" of the total energy operator, projective measurements of a system must conserve energy as defined by the system's energy operator. This paper explores the ideas behind the WAY theorem in hopes of uncovering the origin of the contradiction. After Araki and Yanase published their proof of the WAY theorem, Yanase appended a new condition now known as the Yanase condition. Under the simplifying assumption that the observable being measured has discrete and non-degenerate eigenvalues, we prove that the Yanase condition actually follows from the hypotheses of the original WAY theorem. The paper also proves that the hypotheses of the WAY theorem, together with the simplifying assumption, imply that the energy operator for the measuring apparatus must be a multiple of the identity, which seems physically unlikely. It seems probable that this surprising conclusion, along with the Yanase condition, also holds without the simplifying assumption.