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Rényi squashed entanglement, discord, and relative entropy differences

Seshadreesan, Kaushik P. and Berta, Mario and Wilde, Mark M. (2015) Rényi squashed entanglement, discord, and relative entropy differences. Journal of Physics A: Mathematical and Theoretical, 48 (39). Art. No. 395303. ISSN 1751-8113. https://resolver.caltech.edu/CaltechAUTHORS:20151019-144153297

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Abstract

The squashed entanglement quantifies the amount of entanglement in a bipartite quantum state, and it satisfies all of the axioms desired for an entanglement measure. The quantum discord is a measure of quantum correlations that are different from those due to entanglement. What these two measures have in common is that they are both based upon the conditional quantum mutual information. In Berta et al (2015 J. Math. Phys. 56 022205), we recently proposed Rényi generalizations of the conditional quantum mutual information of a tripartite state on ABC (with C being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system B (with it being left open to show that the Rényi quantity is monotone with respect to local operations on system A). Here we define a Rényi squashed entanglement and a Rényi quantum discord based on a Rényi conditional quantum mutual information and investigate these quantities in detail. Taking as a conjecture that the Rényi conditional quantum mutual information is monotone with respect to local operations on both systems A and B, we prove that the Rényi squashed entanglement and the Rényi quantum discord satisfy many of the properties of the respective original von Neumann entropy based quantities. In our prior work (Berta et al 2015 Phys. Rev. A 91 022333), we also detailed a procedure to obtain Rényi generalizations of any quantum information measure that is equal to a linear combination of von Neumann entropies with coefficients chosen from the set {-1, 0, 1}. Here, we extend this procedure to include differences of relative entropies. Using the extended procedure and a conjectured monotonicity of the Rényi generalizations in the Rényi parameter, we discuss potential remainder terms for well known inequalities such as monotonicity of the relative entropy, joint convexity of the relative entropy, and the Holevo bound.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1088/1751-8113/48/39/395303DOIArticle
http://iopscience.iop.org/article/10.1088/1751-8113/48/39/395303/metaPublisherArticle
http://arxiv.org/abs/1410.1443arXivDiscussion Paper
ORCID:
AuthorORCID
Berta, Mario0000-0002-0428-3429
Additional Information:© 2015 IOP Publishing Ltd. Received 1 June 2015, revised 9 August 2015; Accepted for publication 10 August 2015; Published 14 September 2015. We thank ARP Rau for many helpful discussions about this work and John Calsamiglia for discussions about Rényi quantum discord. KS and MMW are grateful to Naresh Sharma for hosting them for a research visit to the Tata Institute of Fundamental Research during June 2014, where some of the results in this paper were established. MMW is grateful to the Institute for Quantum Information and Matter at Caltech for hospitality during a research visit in July 2014. KS acknowledges support from NSF Grant No. CCF-1350397, the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019, and the Graduate School of Louisiana State University for the 2014–2015 Dissertation Year Fellowship. MMW acknowledges support from the APS-IUSSTF Professorship Awards in Physics, 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 W31P4Q-12-1-0019.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
NSFCCF-1350397
Army Research Office (ARO)W31P4Q-12-1-0019
Louisiana State UniversityUNSPECIFIED
NSFCCF-1350397
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
American Physical SocietyUNSPECIFIED
Issue or Number:39
Record Number:CaltechAUTHORS:20151019-144153297
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20151019-144153297
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:61288
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Oct 2015 21:52
Last Modified:03 Oct 2019 09:06

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