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arXiv:1001.3376

Generalized Cramer-Rao relations for non-relativistic quantum systems

J. S. Dehesa, Angelo Plastino, Pablo Sánchez-Moreno, Christophe Vignat

Published Jan 19, 2010• 9 citations
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Abstract

The Cramer-Rao product of the Fisher information and the variance of a probability density ρ(x), defined on a domain Δ\in R^D, is found to have a minimum value reached by the density associated with the ground state of the harmonic oscillator in Δ, when Δis an unbounded domain. If Δis bounded, the minimum value of the Fisher information is achieved by the ground state of the quantum box described itself by this domain.

DOI
10.48550/arxiv.1001.3376

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The Cramer-Rao product of the Fisher information and the variance of a probability density ρ(x), defined on a domain Δ\in R^D, is found to have a minimum value reached by the density associated with the ground state of the harmonic oscillator in Δ, when Δis an unbounded domain. If Δis bounded, the minimum value of the Fisher information is achieved by the ground state of the quantum box described itself by this domain.
In quantum theory, there are various possibilities of defining logarithmic-derivative operators, each giving rise to a different Likelihood operator, and also to quantum Fisher information (QFI). In each case, the Cramér–Rao (CR) bound is derived from the first principles. It is also noted that if one does not make (often tacitly made) an assumption that the density operator [Formula: see text] and its derivative commute, then they lead to different QFI. Applications to one-mode coherent states (an infinite-dimensional quantum system) and to two-level quantum systems are studied.
Abstract The quantum Cramér–Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér–Rao bound in the phase estimation of a solid-state spin system, provided by a nitrogen-vacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted from coherent dynamical responses of the system under weak parametric modulations, without performing any quantum-state tomography. While optimal parameter estimation has already been observed for quantum devices involving a limited number of degrees of freedom, our method offers a versatile and powerful experimental tool to explore the Cramér–Rao bound and the quantum Fisher information in systems of higher complexity, as relevant for quantum technologies.
Fisher information plays a central role in statistics and quantum metrology, providing the basis for the celebrated Cramér-Rao bound. In this work, we introduce a new information measure based on higher-order Fisher information and show that it naturally leads to a generalized uncertainty relation for parameter estimation, which can be regarded as an extension of the Cramér-Rao bound. As an application, we analyze the case of quantum phase estimation with a single qubit and compare our theoretical bounds with the well-known established hierarchical bounds. Finally, we experimentally validate the proposed framework using a photonic platform.
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The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be understood as a result of extended regimes of optimization. Two experimentally feasible examples are presented to illustrate the theory, saturation of these bounds and estimation precisions beyond the Heisenberg limit are predicted and discussed. A setup to measure non-Hermitian observable is also proposed.