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International Conference on Quantum Information

Knowledge and ignorance in quantum state estimation

Z. Hradil, Jaroslav Řeháček

Published Jan 1, 2011• 0 citations
Publisher

Abstract

An overview of quantum reconstruction methods based on statistical interpretation is presented. In realistic experiments there is always a part of missing information yielding additional ambiguities. Robust estimation strategy based on maximum entropy and maximum likelihood estimation gives the optimal trade-off between ignorance and knowledge.

DOI
10.1364/icqi.2011.qtuc3

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An overview of quantum reconstruction methods based on statistical interpretation is presented. In realistic experiments there is always a part of missing information yielding additional ambiguities. Robust estimation strategy based on maximum entropy and maximum likelihood estimation gives the optimal trade-off between ignorance and knowledge.
Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.
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.
The maximum likelihood strategy for the estimation of group parameters allows one to derive in a general fashion optimal measurements, optimal signal states, and their relations with other information theoretical quantities. These results provide deep insight into the general structure underlying optimal quantum estimation strategies. The entanglement between representation spaces and multiplicity spaces of the group action appears to be the unique kind of entanglement which is really useful for the optimal estimation of group parameters.