[{"abstract":[{"lang":"eng","text":"Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments."}],"doi":"10.1016/j.laa.2022.09.001","day":"01","isi":1,"external_id":{"isi":["000860689600014"]},"date_updated":"2023-08-04T09:24:51Z","year":"2022","citation":{"chicago":"Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s Concavity Theorem and Related Inequalities.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">https://doi.org/10.1016/j.laa.2022.09.001</a>.","ieee":"E. A. Carlen and H. Zhang, “Monotonicity versions of Epstein’s concavity theorem and related inequalities,” <i>Linear Algebra and its Applications</i>, vol. 654. Elsevier, pp. 289–310, 2022.","ama":"Carlen EA, Zhang H. Monotonicity versions of Epstein’s concavity theorem and related inequalities. <i>Linear Algebra and its Applications</i>. 2022;654:289-310. doi:<a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">10.1016/j.laa.2022.09.001</a>","apa":"Carlen, E. A., &#38; Zhang, H. (2022). Monotonicity versions of Epstein’s concavity theorem and related inequalities. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">https://doi.org/10.1016/j.laa.2022.09.001</a>","ista":"Carlen EA, Zhang H. 2022. Monotonicity versions of Epstein’s concavity theorem and related inequalities. Linear Algebra and its Applications. 654, 289–310.","mla":"Carlen, Eric A., and Haonan Zhang. “Monotonicity Versions of Epstein’s Concavity Theorem and Related Inequalities.” <i>Linear Algebra and Its Applications</i>, vol. 654, Elsevier, 2022, pp. 289–310, doi:<a href=\"https://doi.org/10.1016/j.laa.2022.09.001\">10.1016/j.laa.2022.09.001</a>.","short":"E.A. Carlen, H. Zhang, Linear Algebra and Its Applications 654 (2022) 289–310."},"ddc":["510"],"volume":654,"acknowledgement":"Work partially supported by the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337.","title":"Monotonicity versions of Epstein's concavity theorem and related inequalities","intvolume":"       654","publication_status":"published","date_created":"2023-01-16T09:46:38Z","department":[{"_id":"JaMa"}],"article_processing_charge":"Yes (via OA deal)","author":[{"last_name":"Carlen","first_name":"Eric A.","full_name":"Carlen, Eric A."},{"full_name":"Zhang, Haonan","last_name":"Zhang","first_name":"Haonan","id":"D8F41E38-9E66-11E9-A9E2-65C2E5697425"}],"_id":"12216","scopus_import":"1","license":"https://creativecommons.org/licenses/by/4.0/","article_type":"original","publisher":"Elsevier","file_date_updated":"2023-01-27T08:08:39Z","page":"289-310","quality_controlled":"1","oa":1,"publication_identifier":{"issn":["0024-3795"]},"date_published":"2022-12-01T00:00:00Z","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","status":"public","file":[{"date_updated":"2023-01-27T08:08:39Z","file_name":"2022_LinearAlgebra_Carlen.pdf","content_type":"application/pdf","date_created":"2023-01-27T08:08:39Z","file_size":441184,"checksum":"cf3cb7e7e34baa967849f01d8f0c1ae4","file_id":"12415","creator":"dernst","success":1,"access_level":"open_access","relation":"main_file"}],"month":"12","oa_version":"Published Version","project":[{"_id":"eb958bca-77a9-11ec-83b8-c565cb50d8d6","grant_number":"M03337","name":"Curvature-dimension in noncommutative analysis"}],"publication":"Linear Algebra and its Applications","has_accepted_license":"1","language":[{"iso":"eng"}],"keyword":["Discrete Mathematics and Combinatorics","Geometry and Topology","Numerical Analysis","Algebra and Number Theory"]},{"_id":"8373","author":[{"last_name":"Pitrik","first_name":"József","full_name":"Pitrik, József"},{"first_name":"Daniel","last_name":"Virosztek","orcid":"0000-0003-1109-5511","full_name":"Virosztek, Daniel","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87"}],"publication_status":"published","date_created":"2020-09-11T08:35:50Z","department":[{"_id":"LaEr"}],"article_processing_charge":"No","title":"A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means","intvolume":"       609","page":"203-217","ec_funded":1,"quality_controlled":"1","publisher":"Elsevier","article_type":"original","date_updated":"2023-08-04T10:58:14Z","year":"2021","citation":{"ama":"Pitrik J, Virosztek D. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra and its Applications</i>. 2021;609:203-217. doi:<a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">10.1016/j.laa.2020.09.007</a>","apa":"Pitrik, J., &#38; Virosztek, D. (2021). A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. <i>Linear Algebra and Its Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">https://doi.org/10.1016/j.laa.2020.09.007</a>","chicago":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” <i>Linear Algebra and Its Applications</i>. Elsevier, 2021. <a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">https://doi.org/10.1016/j.laa.2020.09.007</a>.","ieee":"J. Pitrik and D. Virosztek, “A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means,” <i>Linear Algebra and its Applications</i>, vol. 609. Elsevier, pp. 203–217, 2021.","mla":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” <i>Linear Algebra and Its Applications</i>, vol. 609, Elsevier, 2021, pp. 203–17, doi:<a href=\"https://doi.org/10.1016/j.laa.2020.09.007\">10.1016/j.laa.2020.09.007</a>.","short":"J. Pitrik, D. Virosztek, Linear Algebra and Its Applications 609 (2021) 203–217.","ista":"Pitrik J, Virosztek D. 2021. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear Algebra and its Applications. 609, 203–217."},"isi":1,"external_id":{"isi":["000581730500011"],"arxiv":["2002.11678"]},"arxiv":1,"doi":"10.1016/j.laa.2020.09.007","day":"15","abstract":[{"text":"It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.","lang":"eng"}],"acknowledgement":"The authors are grateful to Milán Mosonyi for fruitful discussions on the topic, and to the anonymous referee for his/her comments and suggestions.\r\nJ. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum Information Theory, No. 96 141, and by Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K119442, no. K124152, and no. KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01), by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","volume":609,"publication":"Linear Algebra and its Applications","oa_version":"Preprint","project":[{"name":"Geometric study of Wasserstein spaces and free probability","grant_number":"846294","_id":"26A455A6-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"month":"01","language":[{"iso":"eng"}],"keyword":["Kubo-Ando mean","weighted multivariate mean","barycenter"],"date_published":"2021-01-15T00:00:00Z","type":"journal_article","publication_identifier":{"issn":["0024-3795"]},"oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/2002.11678","open_access":"1"}],"status":"public","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8"}]