A Framework for Non-Asymptotic Quantum Information Theory

Tomamichel, M

A Framework for Non-Asymptotic Quantum Information Theory

Tomamichel, M

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Publication Type:: Thesis
Citation:: 2012
Issue Date:: 2012-03-10

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Field	Value	Language
dc.contributor.author	Tomamichel, M https://orcid.org/0000-0001-5410-3329
dc.date.accessioned	2023-03-14T01:31:10Z
dc.date.available	2023-03-14T01:31:10Z
dc.date.issued	2012-03-10
dc.identifier.citation	2012
dc.identifier.uri	http://hdl.handle.net/10453/167285
dc.description.abstract	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. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.
dc.language	en
dc.rights	info:eu-repo/semantics/closedAccess
dc.title	A Framework for Non-Asymptotic Quantum Information Theory
dc.type	Thesis
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
utslib.copyright.status	closed_access	*
dc.date.updated	2023-03-14T01:31:07Z

Abstract:

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. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.

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http://hdl.handle.net/10453/167285