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Quantum information and statistical inference

Alessandra Luati

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The chapter presents some recent results on the relations between classical expected Fisher information and quantum counterparts such as Helstrom information and Wigner-Yanase skew information. The results have relevant application in the evaluation of the efficiency of an unbiased estimator of the parameter of interest (that one that specifies the state of the system) since they lead to approximate quantum Cramér-Rao bounds.

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Generalized Wigner-Yanase skew information and generalized Fisher information

研二郎 柳, Shigeru Furuichi, 憲 栗山

International Symposium on Information Theory and its Applications(ISITA2008), Auckland, New Zealand, 7-10, December, 2008 -- (Abstract)We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cram´er-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.

•Dec 1, 2008•0 citations
The chapter presents some recent results on the relations between classical expected Fisher information and quantum counterparts such as Helstrom information and Wigner-Yanase skew information. The results have relevant application in the evaluation of the efficiency of an unbiased estimator of the parameter of interest (that one that specifies the state of the system) since they lead to approximate quantum Cramér-Rao bounds.
A closed-form expression for Wigner–Yanase skew information in mixed-state quantum systems is derived. It is shown that limit values of the mixing coefficients exist such that Wigner–Yanase information is equal to Helstrom information. The latter constitutes an upper bound for the classical expected Fisher information, hence the inverse Wigner–Yanase information provides an approximate lower bound to the variance of an unbiased estimator of the parameter of interest. The advantage of approximating Helstrom’s sharp bound lies in the fact that Wigner–Yanase information is straightforward to compute, while it is often very difficult to obtain a feasible expression for Helstrom information. In fact, the latter requires the solution of an implicit second order matrix differential equation, while the former requires just scalar differentiation.
We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cramer-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.
The Wigner-Araki-Yanase theorem puts a limitation on the measurement of observables in the presence of a conserved quantity, and the notion of Wigner-Yanase skew information quantifies the amount of information on the values of observables not commuting with the conserved quantity. We demonstrate that the statistical idea underlying the skew information is the Fisher information in the theory of statistical estimation. A quantum Cramér-Rao inequality and a new uncertainty relation in terms of the skew information are established, which shed considerable new light on the relationships between quantum measurement and statistical inference. The result is applied to estimating the evolution speed of quantum states.
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.
As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.
International Symposium on Information Theory and its Applications(ISITA2008), Auckland, New Zealand, 7-10, December, 2008 -- (Abstract)We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cram´er-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.