**Optimal estimation of the overlap between two arbitrary quantum states** - Michalis Skotiniotis

Determining the overlap, , between two arbitrary quantum states, is a fundamental primitive in quantum information processing with applications ranging from quantum fingerprinting and entanglement estimation, to quantum machine learning. Hitherto, the standard protocol for determining the overlap between two quantum states is the so-called SWAP test; given a copy of and the probability of projecting the joint state on its symmetric or antisymmetric part is determined by the overlap of the states. By repeating this measurement on several pairs of copies one can obtain a good estimate of this probability, and hence the overlap. We show that a more precise estimate can be obtained by allowing for general collective measurements on all copies. We derive the statistics of the optimal measurement and compute the optimal mean square error in the asymptotic pointwise and finite Bayesian estimation settings. We also consider two strategies relying on the estimation of one or both the states, and show that, although they are suboptimal, they outperform the SWAP test. As a bonus, we show that the optimal measurement is less invasive than the SWAP test and study the robustness to depolarizing noise for qubit states. This work is in collaboration with M. Fanizza, M. Rosati, J. Calsamiglia and V. Giovannetti. c = tr(ρσ) ρ, σ ρ σ

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