  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 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.
