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Rényi generalizations of the conditional quantum mutual information

Berta, Mario and Seshadreesan, Kaushik P. and Wilde, Mark M. (2015) Rényi generalizations of the conditional quantum mutual information. Journal of Mathematical Physics, 56 (2). Art. No. 022205. ISSN 0022-2488. http://resolver.caltech.edu/CaltechAUTHORS:20150409-090246974

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Abstract

The conditional quantum mutual information I(A; B|C) of a tripartite state ρABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations I α (A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1063/1.4908102 DOIArticle
http://scitation.aip.org/content/aip/journal/jmp/56/2/10.1063/1.4908102PublisherArticle
http://arxiv.org/abs/1403.6102arXivDiscussion Paper
ORCID:
AuthorORCID
Berta, Mario0000-0002-0428-3429
Additional Information:© 2015 AIP Publishing LLC. Received 27 May 2014; accepted 3 February 2015; published online 23 February 2015. K.S. acknowledges support from the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019 and the Graduate school, Louisiana State University. M.M.W. is grateful to the Institute for Quantum Information and Matter at Caltech for hospitality during a research visit in July 2014. M.M.W. acknowledges startup funds from the Department of Physics and Astronomy at LSU, support from the NSF under Award No. CCF-1350397, and support from the DARPA Quiness Program through US Army Research Office Award No. W31P4Q-12-1-0019.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Army Research Office (ARO)W31P4Q-12-1-0019
Louisiana State UniversityUNSPECIFIED
NSFCCF-1350397
Record Number:CaltechAUTHORS:20150409-090246974
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20150409-090246974
Official Citation:Rényi generalizations of the conditional quantum mutual information Berta, Mario and Seshadreesan, Kaushik P. and Wilde, Mark M., Journal of Mathematical Physics, 56, 022205 (2015), DOI:http://dx.doi.org/10.1063/1.4908102
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:56509
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:09 Apr 2015 19:04
Last Modified:17 Aug 2017 00:00

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