  This thesis includes a survey of the results known for private and
approximate private quantum channels. We develop the best known upper bound for
$\epsilon$-randomizing maps, $n+2\log(1/\epsilon)+c$ bits required to
$\epsilon$-randomize an arbitrary $n$-qubit state by improving a scheme of
Ambainis and Smith \cite{AS04} based on small bias spaces \cite{NN90, AGHP92}.
We show by a probabilistic argument that in fact the great majority of random
schemes using slightly more than this many bits of key are also
$\epsilon$-randomizing. We provide the first known non-trivial lower bound for
$\epsilon$-randomizing maps, and develop several conditions on them which we
hope may be useful in proving stronger lower bounds in the future.
