Nondisturbing Quantum Measurement Models

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.