{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T00:09:26Z","timestamp":1774051766011,"version":"3.50.1"},"reference-count":44,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,1,26]],"date-time":"2021-01-26T00:00:00Z","timestamp":1611619200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"e French National Research Agency","award":["ANR-18-CE47-0011"],"award-info":[{"award-number":["ANR-18-CE47-0011"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>We introduce a new quantum R\u00e9nyi divergence<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msubsup><mml:mi>D<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">#<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:math>for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><mml:mo>\u2208<\/mml:mo><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi mathvariant=\"normal\">\u221e<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum R\u00e9nyi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math>-R\u00e9nyi divergence between quantum channels for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>. Second it allows us to prove a chain rule property for the sandwiched<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math>-R\u00e9nyi divergence for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.<\/jats:p>","DOI":"10.22331\/q-2021-01-26-387","type":"journal-article","created":{"date-parts":[[2021,1,26]],"date-time":"2021-01-26T18:52:05Z","timestamp":1611687125000},"page":"387","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":30,"title":["Defining quantum divergences via convex optimization"],"prefix":"10.22331","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6026-4102","authenticated-orcid":false,"given":"Hamza","family":"Fawzi","sequence":"first","affiliation":[{"name":"DAMTP, University of Cambridge, United Kingdom"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8491-0359","authenticated-orcid":false,"given":"Omar","family":"Fawzi","sequence":"additional","affiliation":[{"name":"Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2021,1,26]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"WN Anderson, Jr and GE Trapp. Shorted operators. 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