  We addressed the question of optimality of private quantum channels. We have
shown that the Shannon entropy of the classical key necessary to securely
transfer the quantum information is lower bounded by the entropy exchange of
the private quantum channel $\cal E$ and von Neumann entropy of the ciphertext
state $\varrho^{(0)}$. Based on these bounds we have shown that decomposition
of private quantum channels into orthogonal unitaries (if exists) is optimizing
the entropy. For non-ancillary single qubit PQC we have derived the optimal
entropy for arbitrary set of plaintexts. In particular, we have shown that
except when the (closure of the) set of plaintexts contains all states, one bit
key is sufficient. We characterized and analyzed all the possible single qubit
private quantum channels for arbitrary set of plaintexts. For the set of
plaintexts consisting of all qubit states we have characterized all possible
approximate private quantum channels and we have derived the relation between
the security parameter and the corresponding minimal entropy.
