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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.