Optimal axis compensation in quantum key distribution protocols over unital channels

The axis compensation is a procedure in which the sender and the receiver compensate the axes of their transmitter and detector so that the bit sequence can be transmitted more reliably. We show the optimal axis compensations maximizing the key generation rate for unital channels. We consider the case in which only Bob is allowed to compensate his axis, and the case in which both Alice and Bob are allowed to compensate their axes. In the former case, we show that we should utilize the mismatched measurement outcomes in the channel estimation phase. In the latter case, we show that we do not have to utilize the mismatched measurement outcomes in the channel estimation phase.