{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T20:55:38Z","timestamp":1792788938337,"version":"4.5.0"},"reference-count":22,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T00:00:00Z","timestamp":1694044800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"MTA R\u00e9nyi \u201cLend\u00fclet\u201d","award":["Groups and Graphs Research Group"],"award-info":[{"award-number":["Groups and Graphs Research Group"]}]},{"DOI":"10.13039\/501100011019","name":"NKFIH","doi-asserted-by":"crossref","award":["K 124152 and KKP 139502"],"award-info":[{"award-number":["K 124152 and KKP 139502"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Integral representations of quantum relative entropy, and of the directional second and higher order derivatives of von Neumann entropy, are established, and used to give simple proofs of fundamental, known data processing inequalities: the Holevo bound on the quantity of information transmitted by a quantum communication channel, and, much more generally, the monotonicity of quantum relative entropy under trace-preserving positive linear maps \u2013 complete positivity of the map need not be assumed. The latter result was first proved by M\u00fcller-Hermes and Reeb, based on work of Beigi. For a simple application of such monotonicities, we consider any `divergence&amp;apos; that is non-increasing under quantum measurements, such as the concavity of von Neumann entropy, or various known quantum divergences. An elegant argument due to Hiai, Ohya, and Tsukada is used to show that the infimum of such a `divergence&amp;apos; on pairs of quantum states with prescribed trace distance is the same as the corresponding infimum on pairs of binary classical states. Applications of the new integral formulae to the general probabilistic model of information theory, and a related integral formula for the classical R\u00e9nyi divergence, are also discussed.<\/jats:p>","DOI":"10.22331\/q-2023-09-07-1102","type":"journal-article","created":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T10:27:19Z","timestamp":1694082439000},"page":"1102","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":17,"title":["Integral formula for quantum relative entropy implies data processing inequality"],"prefix":"10.22331","volume":"7","author":[{"given":"P\u00e9ter E.","family":"Frenkel","sequence":"first","affiliation":[{"name":"E\u00f6tv\u00f6s University, Institute of Mathematics, P\u00e1zm\u00e1ny P\u00e9ter s\u00e9t\u00e1ny 1\/C, Budapest, 1117 Hungary"},{"name":"R\u00e9nyi Institute, Budapest, Re\u00e1ltanoda u. 13-15, 1053 Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2023,9,7]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"S. 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