Born's rule and measurement

TL;DR

This paper provides a comprehensive, first-principles derivation of the generalized Born's rule for quantum measurements using POVMs, highlighting its relevance for realistic experimental scenarios and proposing a new pedagogical approach.

Contribution

It offers a self-contained, deductive introduction to quantum measurement and Born's rule in its generalized form, incorporating an intuitive detector definition and historical context.

Findings

Derivation of Born's rule for POVMs from first principles

Inclusion of realistic measurement effects like losses and dark counts

Proposes a new approach for teaching quantum mechanics

Abstract

Born's rule in its conventional textbook form applies to the small class of projective measurements only. It is well-known that a generalization of Born's rule to realistic experiments must be phrased in terms of positive operator valued measures (POVMs). This generalization accounts for things like losses, imperfect measurements, limited detection accuracy, dark detector counts, and the simultaneous measurement of position and momentum. Starting from first principles, this paper gives a self-contained, deductive introduction to quantum measurement and Born's rule, in its generalized form that applies to the results of measurements described by POVMs. It is based on a suggestive definition of what constitutes a detector, assuming an intuitive informal notion of response. The formal exposition is embedded into the context of a variaety of quotes from the literature illuminating historical aspects of the subject. The material presented suggests a new approach to introductory courses on quantum mechanics.