Abstract
We develop priors for Bayes estimation of quantum states that provide minimax state estimation. The relative entropy from the true density operator to a predictive density operator is adopted as a loss function. The proposed prior maximizes the conditional Holevo mutual information, and it is a quantum version of the latent information prior in classical statistics. For one qubit system, we provide a class of measurements that is optimal from the viewpoint of minimax state estimation.
Highlights
In quantum mechanics, the outcome of a measurement is subject to a probability distribution determined from the quantum state of the measured system and the measurement performed.The task of estimating the quantum state from the outcome of measurement is called the quantum estimation and it is a fundamental problem in quantum statistics [1,2,3]
Tanaka and Komaki [4] and Tanaka [5] discussed quantum estimation using the framework of statistical decision theory and showed that Bayesian methods provide better estimation than the maximum likelihood method
We provide a quantum version of the latent information priors and prove that they provide minimax predictive density operators
Summary
The outcome of a measurement is subject to a probability distribution determined from the quantum state of the measured system and the measurement performed. The Bayesian predictive densities based on the latent information priors are minimax under the Kullback–Leibler risk: max R(θ, pπ ) = min max R(θ, p). Note that the fidelity and the trace norm correspond to the Hellinger distance and the total variation distance in the classical statistics, respectively Under this setting, Tanaka and Komaki [4] proved that the Bayesian predictive density operators minimize the Bayes risk: R(θ, σπY )dπ (θ ). Tanaka and Komaki [4] proved that the Bayesian predictive density operators minimize the Bayes risk: R(θ, σπY )dπ (θ ) This is a quantum version of Equation (4). We provide a quantum version of the latent information priors and prove that they provide minimax predictive density operators. These measurements and latent information priors provide robust quantum estimation
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