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arXiv:quant-ph/0602072

Generalized Bayesian predictive density operators

Fuyuhiko Tanaka

Published Feb 7, 2006• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0602072

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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.
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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.
In the quantum state tomography, which is the task of estimating the unknown quantum state on a Hilbert space from measurement data, the standard Bayesian technique is often used.As is well known in Bayesian statistics, it is a nontrivial problem to define a noninformative prior in gen- eral statistical models.Although some definitions for statistical models of mixed states are given by straightforward extension of classical results, there is no counterpart for those of pure states.Recently the author proposed a conceptual extension of the famous reference prior by Bemardo and defined noninformative priors for statistical models of pure states.While the formal extension of the reference prior corresponds to the prior maximizing the von Neumann entropy, our proposed prior is shown to maximize the limit of the quantum Renyi entropy.We show some relations between two principles.
We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on nnn observed copies. Performance is evaluated by quantum relative entropy. For a fixed measurement, the Bayes predictive state is the posterior mean and the optimized conditional loss is its entropy. We then optimize the measurement over all POVMs on the symmetric subspace. For every nonzero positive effect EEE, the corresponding posterior predictive state is μE=(I+nρE)/(n+d)μ_E=(I+nρ_E)/(n+d)μE​=(I+nρE​)/(n+d), where ρEρ_EρE​ is the normalized one-particle marginal of EEE. Since a pure spectrum majorizes every density-operator spectrum, this identity gives an outcome-wise entropy lower bound. Coherent rank-one effects attain the bound, and their Haar orbit yields the highest-weight covariant POVM. Hence this POVM is globally Bayes optimal over all collective measurements and, by covariance, globally minimax. Its exact risk is hd((n+1)/(n+d))h_d((n+1)/(n+d))hd​((n+1)/(n+d)), where hd(r)=−rlog⁡r−(1−r)log⁡((1−r)/(d−1))h_d(r)=-r\log r-(1-r)\log((1-r)/(d-1))hd​(r)=−rlogr−(1−r)log((1−r)/(d−1)). The same arbitrary-effect reduction shows that the highest-weight POVM also maximizes the joint overlap between the latent pure state and its posterior predictive state, equivalently the mean posterior purity, with optimum ((n+1)2+d−1)/(n+d)2((n+1)^2+d-1)/(n+d)^2((n+1)2+d−1)/(n+d)2.