A Framework for Non-Asymptotic Quantum Information Theory

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This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography.

• We investigate the conditional min- and max-entropy for quantum states, generalizations of classical R\'enyi entropies.
• We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states.
• We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits.