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Approximate Private Quantum Channels

Paul Dickinson
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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 $ε$-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.

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Paul Dickinson. 2006-11-02. Approximate Private Quantum Channels. https://arxiv.org/abs/quant-ph/0611037

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