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.
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.
We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on
n 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
E, the corresponding posterior predictive state is
μE=(I+nρE)/(n+d), where
ρE is the normalized one-particle marginal of
E. 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)), where
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.
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.
An essential quantity in quantum information theory is the von Neumann entropy which depends entirely on the quantum density operator. Once known, the density operator reveals the statistics of observables in a quantum process, and the corresponding von Neumann entropy yields the full information content. However, the state, or density operator, of a given system may be unknown. Quantum state observers have been proposed to infer the unknown state of a quantum system. In this note, we show (i) that the von Neumann entropy of the state estimate produced by our quantum state observer is exponentially convergent to that of the system’s true state, and (ii) the relative entropy between the system and observer’s state converges exponentially to zero as long as the system starts in a full-rank state.