Approximate Private Quantum Channels
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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, <em>n</em> + 2log(1/ε) + <em>c</em> bits required to ε-randomize an arbitrary <em>n</em>-qubit state by improving a scheme of Ambainis and Smith  based on small bias spaces [16, 3]. 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 nontrivial 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.