  In this note, we characterize the form of an invertible quantum operation,
i.e., a completely positive trace preserving linear transformation (a CPTP map)
whose inverse is also a CPTP map. The precise form of such maps becomes
important in contexts such as self-testing and encryption. We show that these
maps correspond to applying a unitary transformation to the state along with an
ancilla initialized to a fixed state, which may be mixed.
  The characterization of invertible quantum operations implies that one-way
schemes for encrypting quantum states using a classical key may be slightly
more general than the ``private quantum channels'' studied by Ambainis, Mosca,
Tapp and de Wolf (FOCS 2000). Nonetheless, we show that their results, most
notably a lower bound of 2n bits of key to encrypt n quantum bits, extend in a
straightforward manner to the general case.
