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Entropy

Minimax Estimation of Quantum States Based on the Latent Information Priors

Takayuki Koyama, Takeru Matsuda, Fumiyasu Komaki

Published Nov 16, 2017• 5 citations
Publisher

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.

DOI
10.3390/e19110618

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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.
Quantum state estimation has been widely investigated and there are mainly two approaches proposed: One is based on the point estimation of an unknown parameter and the other is based on the Bayesian method. We adopt the relative entropy from the true state to a predictive density operator as a loss function. We consider exchangeable quantum models with an arbitrary chosen measurement and show that Bayesian predictive density operators are the best predictive density operators when we evaluate them by using the average relative entropy based on a prior. This result is a quantum version of Aitchison's result in classical statistics.
We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Komaki et al. hrefhttp://dx.doi.org/10.3390/e19110618textrm[Entropy19,618(2017)]\\href{http://dx.doi.org/10.3390/e19110618}{\\textrm{[Entropy 19, 618 (2017)]}}hrefhttp://dx.doi.org/10.3390/e19110618textrm[Entropy19,618(2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that any covariant measurement is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement is minimax if it is only covariant under a unitary 2-design. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.
We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.
Recently quantum prediction problem was proposed in the Bayesian framework. It is shown that Bayesian predictive density operators are the best predictive density operators when we evaluate them by using the average relative entropy based on a prior.As an illustrative example, we treat the Gaussian states family adopting the Gaussian distribution as a prior and give the Bayesian predictive density operator with the heterodyne measurement fixed. We show that it is better than the plug-in predictive density operator based on the maximum likelihood estimate by calculating each average relative entropy.
Abstract Quantum state estimation (or state tomography) is an indispensable task in quantum information processing. Because full state tomography that determines all elements of the density matrix is computationally demanding, one usually takes the strategy of assuming a certain model of quantum states and identifying the model parameters. However, it is difficult to make a valid assumption given little prior knowledge on a quantum state of interest, and thus we need a reasonable model selection method for quantum state estimation. Actually, in the classical statistical estimation theory, several types of information criteria have been established and widely used in practice for appropriately choosing a classical statistical model. In this study, we propose quantum information criteria for evaluating the quality of the estimated quantum state in terms of the quantum relative entropy, which is a natural quantum analogue of the classical information criterion defined in terms of Kullback–Leibler divergence. In particular, we derive two quantum information criteria depending on the type of an estimator for the quantum relative entropy; one uses the log-likelihood and the other uses the classical shadow. The general role of information criteria is to predict the performance of an estimated model for unseen data, although it is a function of only sampled data; this generalization capability of the proposed quantum information criteria is evaluated in numerical simulations.
Recently the quantum Bayesian prediction problem was formulated by Tanaka and Komaki (2005). It is shown that Bayesian predictive density operators are the best predictive density operators when we evaluate them by using the averaged quantum relative entropy based on a prior distribution. In the present paper, we adopt the quantum alpha-divergence as a wider class of loss function. The generalized Bayesian predictive density operator is defined and shown to be best among all the estimates of the unknown density operator.