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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Quantum phase resolution and phase distribution

Gunnar G. Bjork, Alexei S. Trifonov, Tedros Tsegaye, Jonas Söderholm

Published Dec 1, 1998• 10 citations
Publisher

Abstract

We discuss generation and measurement strategies for maximum-likelihood estimation of small phase shifts. We demonstrate that optimal encoding of small phase shifts and optimal measurements of small phase shifts require different states. We also show that states with identical phase resolution may have rather different Pegg-Barnett phase distributions. Hence, the connection between phase distribution and quantum phase resolution is rather weak.

DOI
10.1088/1355-5111/10/6/004

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We discuss generation and measurement strategies for maximum-likelihood estimation of small phase shifts. We demonstrate that optimal encoding of small phase shifts and optimal measurements of small phase shifts require different states. We also show that states with identical phase resolution may have rather different Pegg-Barnett phase distributions. Hence, the connection between phase distribution and quantum phase resolution is rather weak.
Various phase concepts may be treated as special cases of the maximum likelihood estimation. For example, the discrete Fourier estimation that actually coincides with the operational phase of Noh, Foug\`eres, and Mandel is obtained for continuous Gaussian signals with phase modulated mean. Since signals in quantum theory are discrete, a prediction different from that given by the Gaussian hypothesis should be obtained as the best fit assuming a discrete Poissonian statistics of the signal. Although the Gaussian estimation gives a satisfactory approximation for fitting the phase distribution of almost any state, the optimal phase estimation offers in certain cases a measurably better performance. This has been demonstrated in a neutron-optical experiment.
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.