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Physical Review A

Optimal partial estimation of multiple phases

Ladislav Mišta, Jaromı́r Fiurášek, Radim Filip

Published Jul 12, 2005• 29 citations
Publisher

Abstract

We derive the optimal trade-off between the estimation fidelity and the fidelity of the state after the estimation for a single copy of an unknown pure state of a $d$-level system belonging to a specific subset of the set of all pure states. The set of states to be considered here is formed by all pure states of a $d$-level system produced by $d$-independent phase shifts of some reference state. We also propose a measurement scheme realizing such an optimal partial estimation of multiple phases in which the trade-off between the fidelities can be controlled by the state of the ancilla.

DOI
10.1103/physreva.72.012311

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We derive the optimal trade-off between the estimation fidelity and the fidelity of the state after the estimation for a single copy of an unknown pure state of a ddd-level system belonging to a specific subset of the set of all pure states. The set of states to be considered here is formed by all pure states of a ddd-level system produced by ddd-independent phase shifts of some reference state. We also propose a measurement scheme realizing such an optimal partial estimation of multiple phases in which the trade-off between the fidelities can be controlled by the state of the ancilla.
We derive an analytical formula for the optimal trade-off between the mean estimation fidelity and the mean fidelity of the qubit state after a partial measurement on NNN identically prepared qubits. We also conjecture an analytical expression for the optimal fidelity trade-off in case of a partial measurement on NNN identical copies of a ddd-level system.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a ddd-dimensional unknown pure state with its NNN copies given as input, which was first obtained by Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bru\ss{} and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVMs using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for M(>N)M(>N)M(>N) copies is universal if it is used for NNN copies as input.
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by Bruss and Macchiavello by establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we show optimal estimators with finite POVM exist but not generally universal, where universality means the fidelity is independent of input states. However any optimal estimator for M(>N) copies is universal if it is used for N copies as input.
Given NNN identical copies of the state of a quantum two-level system, we analyze its optimal estimation. We consider two situations: general pure states and (pure) states restricted to lie on the equator of the Bloch sphere. We perform a complete and comprehensive analysis of the optimal schemes based on local measurements, and give results (optimal measurements, maximum fidelity, etc.) for arbitrary NNN, not necessarily large, within the Bayesian framework. We also make a comparative analysis of the asymptotic limit of these results with those derived from a (pointwise) Cram\'er-Rao type of approach. We give explicit schemes based on local measurements and no classical communication that saturate the fidelity bounds of the most general collective schemes.
We analyse the reconstruction of an unknown pure qubit state. We derive the optimal guess that can be inferred from any set of measurements on N identical copies of the system with the fidelity as a figure of merit. We study in detail the estimation process with individual von Neumann measurements and demonstrate that they are very competitive as compared to (complicated) collective measurements. We compute the expressions of the fidelity for large NNN and show that individual measurement schemes can perform optimally in the asymptotic regime.