  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.

Quantum Physics