Generalized conditional expectations for quantum retrodiction and smoothing

TL;DR

This paper introduces a unified formalism for quantum retrodiction and smoothing, generalizing previous methods and providing a robust foundation for quantum inference involving incompatible observables.

Contribution

It proposes a comprehensive formalism that unifies various quantum inference techniques, extending their applicability and theoretical foundation.

Findings

Unifies multiple quantum inference methods into a single formalism

Provides a well-defined distance measure for incompatible observables

Extends the theoretical foundation beyond nondemolition principles

Abstract

The inference of a hidden variable's historical value, based on observations before and after the fact, is a controversial subject in quantum mechanics. Here I address the controversy by proposing a formalism that unifies and generalizes some of the previous proposals for the task, including the quantum minimum-mean-square-error estimators proposed by Ohki, the generalized conditional expectation proposed by Accardi and Cecchini, the quantum smoothing theory proposed by Tsang, the optimal observables for parameter estimation proposed by Personick, Belavkin, and Grishanin, and the weak values proposed by Aharonov, Albert, and Vaidman. The formalism is based on Ohki's suggestion of a distance between two observables in the Heisenberg picture, which remains well defined for incompatible observables and serves as a more general foundation for quantum inference than Belavkin's nondemolition principle.