Approximate Private Quantum Channels
Dickinson, Paul · arXiv (Cornell University) · 2006
This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for $ε$-randomizing maps, $n+2\log(1/ε)+c$ bits required to $ε$-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 $ε$-randomizing. We provide the first known non-trivial lower bound for $ε$-randomizing maps, and develop several conditions on them which we hope may be useful in proving stronger lower bounds in the future.