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arXiv:1102.1998

Fidelity of Physical Measurements

Thomas B. Bahder

Published Feb 9, 2011• 0 citations
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Abstract

The fidelity (Shannon mutual information between measurements and physical quantities) is proposed as a quantitative measure of the quality of physical measurements. The fidelity does not depend on the true value of unknown physical quantities (as does the Fisher information) and it allows for the role of prior information in the measurement process. The fidelity is general enough to allow a natural comparison of the quality of classical and quantum measurements. As an example, the fidelity is used to compare the quality of measurements made by a classical and a quantum Mach-Zehnder interferometer.

DOI
10.48550/arxiv.1102.1998

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The fidelity (Shannon mutual information between measurements and physical quantities) is proposed as a quantitative measure of the quality of physical measurements. The fidelity does not depend on the true value of unknown physical quantities (as does the Fisher information) and it allows for the role of prior information in the measurement process. The fidelity is general enough to allow a natural comparison of the quality of classical and quantum measurements. As an example, the fidelity is used to compare the quality of measurements made by a classical and a quantum Mach-Zehnder interferometer.
For a generic interferometer, the conditional probability density distribution $p(\ensuremath{\phi}\ensuremath{\mid}m)$, for the phase $\ensuremath{\phi}$ given measurement outcome mmm will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕\sqrt{N}$ (the standard limit), or $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity---the fidelity of an interferometer---defined as the Shannon mutual information between the phase shift $\ensuremath{\phi}$ and the measurement outcomes mmm. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. For the case where there exists no a priori information on the phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity than Fock state input, for the same photon number.
Determining the phase in one arm of a quantum interferometer is discussed taking into account the three nonideal aspects in real experiments: nondeterministic state preparation, nonunitary state evolution due to losses during state propagation, and imperfect state detection. A general expression is written for the probability of a measurement outcome taking into account these three nonideal aspects. As an example of applying the formalism, the classical Fisher information and fidelity (Shannon mutual information between phase and measurements) are computed for few-photon Fock and N00N states input into a lossy Mach-Zehnder interferometer. These three nonideal aspects lead to qualitative differences in phase estimation, such as a decrease in fidelity and Fisher information that depends on the true value of the phase.
We present the amount of information, fidelity, and reversibility obtained by arbitrary quantum measurements on completely unknown states. These quantities are expressed as functions of the singular values of a measurement operator corresponding to the obtained outcome. As an example, we consider a class of quantum measurements with highly degenerate singular values to discuss trade-offs among information, fidelity, and reversibility. The trade-offs are at the level of a single outcome, in the sense that the quantities pertain to each single outcome rather than the average over all possible outcomes.
Numerous quantum mechanical schemes have been proposed that are intended to improve the sensitivity to rotation provided by the classical Sagnac effect in gyroscopes. A general metric is needed that can compare the performance of the new quantum systems with the classical systems. The fidelity (Shannon mutual information between the measurement and the rotation rate) is proposed as a metric that is capable of this comparison. A theoretical upper bound is derived for the fidelity of an ideal classical Sagnac gyroscope. This upper bound for the classical Sagnac gyroscope should be used as a benchmark to compare the performance of proposed enhanced classical and quantum rotation sensors. In fact, the fidelity is general enough to compare the quality of two different apparatuses (two different experiments) that attempt to measure the same quantity.