  Private quantum channels are the quantum analogue of the classical one-time
pad. Conditional expectations and trace vectors are notions that have been part
of operator algebra theory for several decades. We show that the theory of
conditional expectations and trace vectors is intimately related to that of
private quantum channels. Specifically we give a new geometric characterization
of single qubit private quantum channels that relies on trace vectors. We
further show that trace vectors completely describe the private states for
quantum channels that are themselves conditional expectations. We also discuss
several examples.
