Quantum measuring processes of continuous observables

TLDR

Recent work by Srinivas and Mercer on continuous observable measurements is contextualized by the paper’s findings. The paper aims to establish a theoretical foundation for measuring continuous quantum observables. The authors generalize von Neumann’s measurement model to continuous observables, linking each process to a completely positive instrument and proving the Wigner–Araki–Yanase theorem within this framework. They show that no weakly repeatable instruments exist for non‑discrete observables, implying that repeatable measurement processes cannot be realized for such observables.

Abstract

The purpose of this paper is to provide a basis of theory of measurements of continuous observables. We generalize von Neumann’s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables, and show how every such measuring process determines the state change caused by the measurement. We establish a one-to-one correspondence between completely positive instruments in the sense of Davies and Lewis and the state changes determined by the measuring processes. We also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics, so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis. A proof of the Wigner–Araki–Yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework. We also discuss the implication of these results for the recent results due to Srinivas and due to Mercer on measurements of continuous observables.