Optimal Covariant Measurement of Momentum on a Half Line in Quantum Mechanics

TL;DR

This paper develops an optimal covariant measurement framework for estimating momentum on a half line in quantum mechanics, overcoming the non-observable nature of the momentum operator in this domain.

Contribution

It introduces a method to construct an optimal covariant POVM for momentum on a half line by extending the Hilbert space and explicitly modeling the measurement process.

Findings

Constructed a self-adjoint momentum operator on a half line.

Explicitly modeled the measurement process using a Hamiltonian.

Derived the optimal covariant POVM for momentum measurement.

Abstract

We cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it self-adjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure (POVM) of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.