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arXiv:quant-ph/0307199 · Physical Review A

Collective versus local measurements in a qubit mixed-state estimation

E. Bagán, M. Baig, R. Muñoz-Tapia, Alberto Rodríguez

Published Jan 23, 2004• 45 citations
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Abstract

We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as $1/N.$ For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as ${1/N}^{1/4},$ while our approach leads to an error proportional to ${1/N}^{3/4}.$

DOI
10.1103/physreva.69.010304

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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.
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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.