{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T19:02:58Z","timestamp":1767034978898},"reference-count":12,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T00:00:00Z","timestamp":1714435200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ERC","award":["337603"],"award-info":[{"award-number":["337603"]}]},{"name":"ERC","award":["648913"],"award-info":[{"award-number":["648913"]}]},{"name":"MINECO","award":["FIS2013-40627-P"],"award-info":[{"award-number":["FIS2013-40627-P"]}]},{"name":"MINECO","award":["FIS2016-86681-P"],"award-info":[{"award-number":["FIS2016-86681-P"]}]},{"name":"MINECO","award":["PID2019-107609GB-I00"],"award-info":[{"award-number":["PID2019-107609GB-I00"]}]},{"DOI":"10.13039\/100000925","name":"John Templeton Foundation","doi-asserted-by":"crossref","award":["48322"],"award-info":[{"award-number":["48322"]}],"id":[{"id":"10.13039\/100000925","id-type":"DOI","asserted-by":"crossref"}]},{"name":"ANR","award":["14-CE25-0033"],"award-info":[{"award-number":["14-CE25-0033"]}]},{"name":"ANR","award":["ANR-11-LABX0040"],"award-info":[{"award-number":["ANR-11-LABX0040"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space A<\/mml:mi><\/mml:mrow><\/mml:math> to states on some output Hilbert space B<\/mml:mi><\/mml:mrow><\/mml:math> can be compressed into one with order d<\/mml:mi>log<\/mml:mi>&#x2061;<\/mml:mo>d<\/mml:mi><\/mml:math> Kraus operators, where d<\/mml:mi>=<\/mml:mo>max<\/mml:mo>(<\/mml:mo>|<\/mml:mo><\/mml:mrow>A<\/mml:mi><\/mml:mrow>|<\/mml:mo><\/mml:mrow>,<\/mml:mo>|<\/mml:mo><\/mml:mrow>B<\/mml:mi><\/mml:mrow>|<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:math>, hence much less than |<\/mml:mo><\/mml:mrow>A<\/mml:mi><\/mml:mrow>|<\/mml:mo><\/mml:mrow>|<\/mml:mo><\/mml:mrow>B<\/mml:mi><\/mml:mrow>|<\/mml:mo><\/mml:mrow><\/mml:math>. In the case where the channel's outputs are all very mixed, this can be improved to order d<\/mml:mi><\/mml:math>. We discuss the optimality of this result as well as some consequences.<\/jats:p>","DOI":"10.22331\/q-2024-04-30-1320","type":"journal-article","created":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T12:48:25Z","timestamp":1714481305000},"page":"1320","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Approximating quantum channels by completely positive maps with small Kraus rank"],"prefix":"10.22331","volume":"8","author":[{"given":"C\u00e9cilia","family":"Lancien","sequence":"first","affiliation":[{"name":"Institut Fourier & CNRS, Universit\u00e9 Grenoble Alpes, 38610 Gi\u00e8res, France"}]},{"given":"Andreas","family":"Winter","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica: Grup d\u2019Informaci\u00f3 Qu\u00e0ntica, Universitat Aut\u00f2noma de Barcelona, 08193 Bellaterra (Barcelona), Spain"},{"name":"Instituci\u00f3 Catalana de Recerca i Estudis Avan\u00e7ats, 08010 Barcelona, Spain"}]}],"member":"9598","published-online":{"date-parts":[[2024,4,30]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Dorit Aharonov, Alexei Kitaev, Noam Nisan. Quantum circuits with mixed states. In Proc. of the 13th Annual ACM Symposium on Theory of Computing (STOC'98), 20\u201330, 1998.","DOI":"10.1145\/276698.276708"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Guillaume Aubrun. On almost randomizing channels with a short Kraus decomposition. Commun. Math. Phys. 288(3):1103\u20131116, 2009.","DOI":"10.1007\/s00220-008-0695-y"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Koenraad M.R. Audenaert. A sharp Fannes-type inequality for the von Neumann entropy. J. Phys. A 40(28):8127\u20138136, 2007.","DOI":"10.1088\/1751-8113\/40\/28\/S18"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Fernando G.S.L. Brand\u00e3o, Aram W. Harrow, Micha\u0142 Horodecki. Local random quantum circuits are approximate polynomial-designs. Commun. Math. Phys. 346(2):397\u2013434, 2016.","DOI":"10.1007\/s00220-016-2706-8"},{"key":"4","unstructured":"Djalil Chafa\u00ef, Olivier Gu\u00e9don, Guillaume Lecu\u00e9, Alain Pajor. Interactions between compressed sensing, random matrices and high dimensional geometry. Panoramas et synth\u00e8se 37, Soci\u00e9t\u00e9 Math\u00e9matique de France, 2012. Pdf available here."},{"key":"5","doi-asserted-by":"publisher","unstructured":"Man-Duen Choi. Completely positive linear maps on complex matrices. Lin. Alg. Appl. 10(3):285\u2013290, 1975.","DOI":"10.1016\/0024-3795(75)90075-0"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Vittorio Giovannetti, Alexander S. Holevo. Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75(046001), 2012.","DOI":"10.1088\/0034-4885\/75\/4\/046001"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Patrick Hayden, Debbie W. Leung, Peter W. Shor, Andreas Winter. Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250(2):371\u2013391, 2004.","DOI":"10.1007\/s00220-004-1087-6"},{"key":"8","doi-asserted-by":"publisher","unstructured":"C\u00e9cilia Lancien, Andreas Winter. Distinguishing multi-partite states by local measurements. Commun. Math. Phys. 323(2):555\u2013573, 2013.","DOI":"10.1007\/s00220-013-1779-x"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Gilles Pisier. The Volume of Convex Bodies and Banach Spaces Geometry. Cambridge Tracts in Mathematics 94, Cambridge University Press, Cambridge, 1989.","DOI":"10.1017\/CBO9780511662454"},{"key":"10","doi-asserted-by":"publisher","unstructured":"William F. Stinespring. Positive functions on C*-algebras. In Proc. of the American Mathematical Society 6(2):211\u2013216, 1955.","DOI":"10.2307\/2032342"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Andreas Winter. Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints. Commun. Math. Phys. 347(1):291\u2013313, 2016.","DOI":"10.1007\/s00220-016-2609-8"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-04-30-1320\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T12:49:34Z","timestamp":1714481374000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2024-04-30-1320\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,30]]},"references-count":12,"URL":"https:\/\/doi.org\/10.22331\/q-2024-04-30-1320","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,30]]},"article-number":"1320"}}