Approximate Private Quantum Channels

Abstract

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.

78 pages, 1 figure. Master's Thesis accepted at University of Waterloo

References in corpus (1)

• Randomizing quantum states: Constructions and applications