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arXiv:1011.2546 · Progress in Informatics

Phase estimation with photon number constraint

Masahito HAYASHI

Published Mar 1, 2011• 19 citations
arXivHTMLPDF

Abstract

Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology.This is because the input noon state provides the maximum Fisher information at the specific point.However, the Fisher information does not necessarily give the attainable bound for estimation error.In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation.We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.

DOI
10.2201/niipi.2011.8.9

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Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology.This is because the input noon state provides the maximum Fisher information at the specific point.However, the Fisher information does not necessarily give the attainable bound for estimation error.In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation.We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.
The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramer-Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form $ $ for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.
The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramér–Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.
When estimating the phase of a single mode, the quantum Fisher information for a pure probe state is proportional to the photon number variance of the probe state.In this work, we point out particular states that offer photon number distributions exhibiting a large variance, which would help to improve the local estimation precision.These theoretical examples are expected to stimulate the community to put more attention to those states that we found, and to work towards their experimental realization and usage in quantum metrology.
Abstract When estimating the phase of a single mode, the quantum Fisher information for a pure probe state is proportional to the photon number variance of the probe state. In this work, we point out particular states that offer photon number distributions exhibiting a large variance, which would help to improve the local estimation precision. These theoretical examples are expected to stimulate the community to put more attention to those states that we found, and to work towards their experimental realization and usage in quantum metrology.
We investigate an optimal distance of two components in an entangled coherent state for quantum phase estimation in lossy interferometry. The optimal distance is obtained by an economical point, representing the quantum Fisher information that we can extract per input energy. Maximizing the formula of the quantum Fisher information over an input mean photon number, we show that, as the losses of the interferometer increase, it can be more beneficial to prepare an initially entangled coherent state that is less entangled. This represents that the optimal distance of the two-mode components decreases with more loss in the interferometry. Under the constraint of the input mean photon number, we obtain that the optimal entangled coherent state is more robust than a separable coherent state, even in a high photon loss rate. The optimal entangled coherent state preserves quantum advantage over the standard interferometric limit of the separable coherent state. We also show that the corresponding optimal measurement requires correlation measurement bases.
Abstract Photon counting measurement has been regarded as the optimal measurement scheme for phase estimation in the squeezed-state interferometry, since the classical Fisher information equals to the quantum fisher information and scales as n ¯ 2 for given input number of photons n ¯ . However, it requires photon-number-resolving detectors with a large enough resolution threshold. Here we show that a collection of N -photon detection events for N up to the resolution threshold ∼ n ¯ can result in the ultimate estimation precision beyond the shot-noise limit. An analytical formula has been derived to obtain the best scaling of the fisher information.