Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means
A regret lower bound is established, showing that the regret rates of Exponential Weights and the algorithm of Tseng et al. (2025) have optimal polynomial dependence on the dimension, which leads to a quantum state tomography algorithm that is sample optimal up to polylogarithmic factors in the sample size.