Limits and restrictions of private quantum channel

We study private quantum channels on a single qubit, which encrypt given set of plaintext states $P$. Specifically, we determine all achievable states $\rho^{(0)}$ (average output of encryption) and for each particular set $P$ we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set $P$ is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional $P$ varies continuously between 1 and 2 bits depending on the state $\rho^{(0)}$. Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.