New Characterizations of Matrix $Φ$-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity

Cheng, H-C; Hsieh, M-H

New Characterizations of Matrix $Φ$-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity

Cheng, H-C Hsieh, M-H

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Publication Type:: Working Paper
Citation:: 2015
Issue Date:: 2015

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Field	Value	Language
dc.contributor.author	Cheng, H-C
dc.contributor.author	Hsieh, M-H
dc.date.accessioned	2023-03-13T23:12:09Z
dc.date.available	2023-03-13T23:12:09Z
dc.date.issued	2015
dc.identifier.citation	2015
dc.identifier.uri	http://hdl.handle.net/10453/167239
dc.description.abstract	We derive new characterizations of the matrix $\Phi$-entropies introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014}]. These characterizations help to better understand the properties of matrix $\Phi$-entropies, and are a powerful tool for establishing matrix concentration inequalities for matrix-valued functions of independent random variables. In particular, we use the subadditivity property to prove a Poincar\'e inequality for the matrix $\Phi$-entropies. We also provide a new proof for the matrix Efron-Stein inequality. Furthermore, we derive logarithmic Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and with Gaussian distributions. Our proof relies on the powerful matrix Bonami-Beckner inequality. Finally, the Holevo quantity in quantum information theory is closely related to the matrix $\Phi$-entropies. This allows us to upper bound the Holevo quantity of a classical-quantum ensemble that undergoes a special Markov evolution.
dc.language	en
dc.relation	http://purl.org/au-research/grants/arc/FT140100574
dc.rights	info:eu-repo/semantics/openAccess
dc.subject	01 Mathematical Sciences, 02 Physical Sciences
dc.subject.classification	Mathematical Physics
dc.title	New Characterizations of Matrix $Φ$-Entropies, Poincaré and Sobolev Inequalities and an Upper Bound to Holevo Quantity
dc.type	Working Paper
utslib.for	01 Mathematical Sciences
utslib.for	02 Physical Sciences
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Students
pubs.organisational-group	/University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
utslib.copyright.status	open_access	*
dc.date.updated	2023-03-13T23:12:08Z

Abstract:

We derive new characterizations of the matrix $\Phi$-entropies introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014}]. These characterizations help to better understand the properties of matrix $\Phi$-entropies, and are a powerful tool for establishing matrix concentration inequalities for matrix-valued functions of independent random variables. In particular, we use the subadditivity property to prove a Poincar\'e inequality for the matrix $\Phi$-entropies. We also provide a new proof for the matrix Efron-Stein inequality. Furthermore, we derive logarithmic Sobolev inequalities for matrix-valued functions defined on Boolean hypercubes and with Gaussian distributions. Our proof relies on the powerful matrix Bonami-Beckner inequality. Finally, the Holevo quantity in quantum information theory is closely related to the matrix $\Phi$-entropies. This allows us to upper bound the Holevo quantity of a classical-quantum ensemble that undergoes a special Markov evolution.

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http://hdl.handle.net/10453/167239