Full Counting Statistics of Non-Commuting Variables: the Case of Spin Counts

TL;DR

This paper investigates the full counting statistics of non-commuting spin variables, revealing how detector back-action influences measurement outcomes and leads to non-Gaussian distributions with large higher cumulants.

Contribution

It introduces a general model for detector dynamics and a path-integral approach to evaluate FCS for non-commuting variables, highlighting the impact of detector back-action.

Findings

FCS depends on detector dynamics and can differ from naive predictions.

The probability distribution exhibits large higher cumulants and non-Gaussian behavior.

A transfer-matrix method is used to evaluate FCS in a diffusive detector model.

Abstract

We discuss the Full Counting Statistics of non-commuting variables with the measurement of successive spin counts in non-collinear directions taken as an example. We show that owing to an irreducible detector back-action, the FCS in this case may be sensitive to the dynamics of the detectors, and may differ from the predictions obtained with using a naive version of the Projection Postulate. We present here a general model of detector dynamics and path-integral approach to the evaluation of FCS. We concentrate further on a simple "diffusive" model of the detector dynamics where the FCS can be evaluated with transfer-matrix method. The resulting probability distribution of spin counts is characterized by anomalously large higher cumulants and substantially deviates from Gaussian Statistics.