Nondisturbing Quantum Measurement Models

TL;DR

This paper develops models for nondisturbing quantum measurements on finite-dimensional systems, providing formulas and examples to simplify understanding of measurement interactions that do not disturb the quantum state.

Contribution

It introduces a framework for nondisturbing measurement models, deriving formulas and illustrating with examples, advancing the theoretical understanding of quantum measurement processes.

Findings

Formulas for observables and instruments in nondisturbing models

Simplification of entanglement in nondisturbing interactions

Examples demonstrating unitary nondisturbing channels

Abstract

A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to $MM$s on finite-dimensional Hilbert spaces. Suppose we want to measure an observable $A$ whose outcomes $A_x$ are represented by positive operators (effects) on a Hilbert Space $H$. We call $H$ the base or object system. We interact $H$ with a probe system on another Hilbert space $K$ by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable $F$ whose outcomes $F_x$ are measured by an apparatus that is frequently (but need not be) classical in practice. The $MM$ protocol gives a method for determining the probability of an outcome $A_x$ for any state of $H$ in terms of the outcome $F_x$. The interaction channel usually entangles this state with an initial probe state of $K$ that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing $MM$s. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.