Exact minimax regret rate for online learning of quantum states

Determine the exact minimax-optimal regret rate for online learning of quantum states with logarithmic loss, including whether the rate is of order [?]

Background

The paper proves that no online learning of quantum states (OLQS) algorithm can achieve a worst-case regret bound of O(d^\gamma \log T) for any \gamma<2, while Exponential Weights and Q-VB-FTRL achieve regret with polynomial dependence O(d^2). However, this lower bound does not match the known upper bound and therefore does not identify the minimax regret rate.

The authors explicitly state that the existing lower bound of \Omega(d\log T), inherited from online portfolio selection, is insufficient to determine whether the minimax regret is \Theta(d^2\log T).