The ensemble Kalman filter (EnKF) is a fundamental data assimilation method widely used across science and engineering, but its performance can deteriorate when the state dimension exceeds the ensemble size. We first investigate the theoretical properties of the EnKF in high-dimensional settings and derive one- and multi-step mean squared error bounds relative to the oracle Kalman filter analysis. Next, we introduce a multi-bandable covariance class and a localization estimator for high-dimensional tensor data. The estimator attains minimax-optimal convergence rates under the spectral and Frobenius norms. Finally, we develop a high-dimensional EnKF (HD-EnKF) that combines localization, inflation, and iterative updates. Numerical experiments with the Lorenz–96 model demonstrate improved assimilation performance under multiple settings.