Ziv-Zakai Error Bounds for Quantum Parameter Estimation

I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cramér-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 Cramér-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 Cramér-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.

Figure 1

Left column: fidelity of a pair of (a) coherent states, (c) rectangle states, and (e) products of Rivas-Luis states with $ϵ=0.1$ and $N_j=1$ with phase difference $\tau$. Right column: mean-square-error lower bounds versus the average photon number on a log-log scale for (b) coherent states, (d) rectangle states, and (f) products of Rivas-Luis states with $ϵ=0.1$, $N_j=1$, and $W=2\pi$; straight lines connecting the numerically calculated points are guides for eyes.