Quantum Estimation via Sequential Measurements

TL;DR

This paper develops a generalized central limit theorem for sequential quantum measurements, showing that data averages become normal and independent of initial states, and demonstrates how correlations can enhance parameter estimation accuracy.

Contribution

It introduces a diagrammatic approach to analyze non-i.i.d. quantum measurement data, extending previous results and exploiting correlations to improve estimation precision.

Findings

Asymptotic normality of measurement data distribution

Correlations among outcomes can be used to improve estimation accuracy

Application to estimating thermal reservoir temperature

Abstract

The problem of estimating a parameter of a quantum system through a series of measurements performed sequentially on a quantum probe is analyzed in the general setting where the underlying statistics is explicitly non-i.i.d. We present a generalization of the central limit theorem in the present context, which under fairly general assumptions shows that as the number $N$ of measurement data increases the probability distribution of functionals of the data (e.g., the average of the data) through which the target parameter is estimated becomes asymptotically normal and independent of the initial state of the probe. At variance with the previous studies [M. Gu\c{t}\u{a}, Phys. Rev. A 83, 062324 (2011); M. van Horssen and M. Gu\c{t}\u{a}, J. Math. Phys. 56, 022109 (2015)] we take a diagrammatic approach, which allows one to compute not only the leading orders in $N$ of the moments of the average of the data but also those of the correlations among subsequent measurement outcomes. In particular our analysis points out that the latter, which are not available in usual i.i.d. data, can be exploited in order to improve the accuracy of the parameter estimation. An explicit application of our scheme is discussed by studying how the temperature of a thermal reservoir can be estimated via sequential measurements on a quantum probe in contact with the reservoir.