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. We consider the case in which only the receiver is allowed to compensate his axis, and the case in which both the sender and the receiver are allowed to compensate their axes. For unital channels, we clarify that the optimal key generation rates for both cases coincide if they utilize the mismatched measurement outcomes in the channel estimation. We also clarify that the optimal key generation rates for both cases do not coincide in general if they do not utilize the mismatched measurement outcomes in the channel estimation.