We investigate the statistical convergence rate of a Bayesian low-rank tensor estimator, and derive the minimax optimal rate for learning a low-rank tensor. Our problem setting is the regression problem where the regression coefficient forms a tensor structure. This problem setting occurs in many practical applications, such as collaborative filtering, multi-task learning, and spatio-temporal data analysis. The convergence rate of the Bayes tensor estimator is analyzed in terms of both in-sample and out-of-sample predictive accuracies. It is shown that a fast learning rate is achieved without any strong convexity of the observation. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, that is, the near optimal rate depending on the true rank is achieved even if it is not known a priori. Finally, we show the minimax optimal learning rate for the tensor estimation problem, and thus show that the derived bound of the Bayes estimator is tight and actually near minimax optimal.