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Optimal estimation of group transformations using entanglement

G. Chiribella·G. M. D'Ariano·M. F. Sacchi
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Abstract

We derive the optimal input states and the optimal quantum measurements for estimating the unitary action of a given symmetry group, showing how the optimal performance is obtained with a suitable use of entanglement. Optimality is defined in a Bayesian sense, as minimization of the average value of a given cost function. We introduce a class of cost functions that generalizes the Holevo class for phase estimation, and show that for states of the optimal form all functions in such a class lead to the same optimal measurement. A first application of the main result is the complete proof of the optimal efficiency in the transmission of a Cartesian reference frame. As a second application, we derive the optimal estimation of a completely unknown two-qubit maximally entangled state, provided that N copies of the state are available. In the limit of large N, the fidelity of the optimal estimation is shown to be 1-3/(4N).

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G. Chiribella, G. M. D'Ariano, M. F. Sacchi. 2005-08-26. Optimal estimation of group transformations using entanglement. https://doi.org/10.1103/physreva.72.042338

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