The quantum Bell-Ziv-Zakai bounds and Heisenberg limits for waveform estimation

TL;DR

This paper introduces quantum Bell-Ziv-Zakai bounds for multiparameter estimation, deriving a generalized Heisenberg limit for measuring stochastic optical phase signals with specific spectral properties.

Contribution

It develops quantum bounds for multiparameter estimation and extends the Heisenberg limit to stochastic optical phase measurements with power-law spectra.

Findings

Mean-square error lower bound scales as 1/N^{2(p-1)/(p+1)}

Sampling and interpolation can achieve the bound for any p>1

Provides a rigorous quantum limit for stochastic phase measurement

Abstract

We propose quantum versions of the Bell-Ziv-Zakai lower bounds on the error in multiparameter estimation. As an application we consider measurement of a time-varying optical phase signal with stationary Gaussian prior statistics and a power law spectrum $\sim 1/|\omega|^p$, with $p>1$. With no other assumptions, we show that the mean-square error has a lower bound scaling as $1/{\cal N}^{2(p-1)/(p+1)}$, where ${\cal N}$ is the time-averaged mean photon flux. Moreover, we show that this accuracy is achievable by sampling and interpolation, for any $p>1$. This bound is thus a rigorous generalization of the Heisenberg limit, for measurement of a single unknown optical phase, to a stochastically varying optical phase.