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arXiv:quant-ph/9609012 · Physical Review A

Quantum-state estimation

Z. Hradil

Published Mar 1, 1997• 542 citations
arXivHTMLPDF

Abstract

An algorithm for quantum-state estimation based on the maximum-likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix.
DOI
10.1103/physreva.55.r1561

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Quantum state estimation is an important task in the realm of experimental quantum information science. While the common approaches to this task are generally adequate, the maximum-likelihood method is frequently used with parameterizations that do not strictly satisfy the statistical requirements of maximum-likelihood estimation. We show that the source of this issue is in the structure of the parameter space and we introduce a density matrix parameterization that provides direct control over this structure. The parameterization leads to a natural quantum state estimation algorithm that does satisfy the statistical requirements of maximum-likelihood estimation. Finally, we examine the algorithm through the analysis of several data sets and experimental results.
Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.