Ziv-Zakai Error Bounds for Quantum Parameter Estimation

TL;DR

This paper introduces quantum Ziv-Zakai bounds as alternative limits for quantum parameter estimation, providing tighter bounds than quantum Cramér-Rao bounds in certain non-Gaussian regimes, especially in optical phase estimation.

Contribution

It develops quantum Ziv-Zakai bounds, offering a new approach that can be tighter than quantum Cramér-Rao bounds for non-Gaussian states in quantum estimation.

Findings

Quantum Ziv-Zakai bounds can be tighter than quantum Cramér-Rao bounds for non-Gaussian states.

The bounds include a 'Heisenberg' limit scaling with average energy.

Application to optical phase estimation demonstrates the bounds' effectiveness.

Abstract

I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cram\'er-Rao bounds for quantum parameter estimation. From a simple form of the proposed bounds, I derive both a "Heisenberg" error limit that scales with the average energy and a limit similar to the quantum Cram\'er-Rao bound that scales with the energy variance. These results are further illustrated by applying the bound to a few examples of optical phase estimation, which show that a quantum Ziv-Zakai bound can be much higher and thus tighter than a quantum Cram\'er-Rao bound for states with highly non-Gaussian photon-number statistics in certain regimes and also stay close to the latter where the latter is expected to be tight.