Physics – Quantum Physics
Scientific paper
2011-11-15
Physics
Quantum Physics
v1: preliminary result, 3 pages; v2: major update, 4 pages + supplementary calculations, v3: another major update, added proof
Scientific paper
In classical statistics, the Ziv-Zakai bounds give lower limits to the mean-square error in a parameter estimation problem by relating it to the error probability in a binary hypothesis testing problem. Compared with the more well known Cram\'er-Rao bounds, the Ziv-Zakai bounds are often much tighter to the achievable error when the likelihood function is highly non-Gaussian and the number of trials is limited. Here I propose quantum versions of the Ziv-Zakai bounds, using the error bounds for quantum state discrimination to derive lower limits to the mean-square error in quantum parameter estimation. A quantum Ziv-Zakai bound is shown to be able to produce 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 illustrated by applying the bound to a few examples of optical phase estimation, which demonstrate the superiority of the quantum Ziv-Zakai bounds over the quantum Cram\'er-Rao bounds in handling states with highly non-Gaussian photon-number statistics.
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