{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T07:38:11Z","timestamp":1747208291940,"version":"3.40.5"},"reference-count":28,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2018,2,5]],"date-time":"2018-02-05T00:00:00Z","timestamp":1517788800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Given a linear map\u03a6<\/mml:mi>:<\/mml:mo>M<\/mml:mi>n<\/mml:mi><\/mml:msub>\u2192<\/mml:mo>M<\/mml:mi>m<\/mml:mi><\/mml:msub><\/mml:math>, its multiplicity maps are defined as the family of linear maps\u03a6<\/mml:mi>\u2297<\/mml:mo>id<\/mml:mtext><\/mml:mrow>k<\/mml:mi><\/mml:mrow><\/mml:msub>:<\/mml:mo>M<\/mml:mi>n<\/mml:mi><\/mml:msub>\u2297<\/mml:mo>M<\/mml:mi>k<\/mml:mi><\/mml:msub>\u2192<\/mml:mo>M<\/mml:mi>m<\/mml:mi><\/mml:msub>\u2297<\/mml:mo>M<\/mml:mi>k<\/mml:mi><\/mml:msub><\/mml:math>, whereid<\/mml:mtext><\/mml:mrow>k<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math>denotes the identity onM<\/mml:mi>k<\/mml:mi><\/mml:msub><\/mml:math>. Let\u2016<\/mml:mo>\u22c5<\/mml:mo>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub><\/mml:math>denote the trace-norm on matrices, as well as the induced trace-norm on linear maps of matrices, i.e.\u2016<\/mml:mo>\u03a6<\/mml:mi>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub>=<\/mml:mo>max<\/mml:mo>{<\/mml:mo>\u2016<\/mml:mo>\u03a6<\/mml:mi>(<\/mml:mo>X<\/mml:mi>)<\/mml:mo>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub>:<\/mml:mo>X<\/mml:mi>\u2208<\/mml:mo>M<\/mml:mi>n<\/mml:mi><\/mml:msub>,<\/mml:mo>\u2016<\/mml:mo>X<\/mml:mi>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub>=<\/mml:mo>1<\/mml:mn>}<\/mml:mo><\/mml:math>. A fact of fundamental importance in both operator algebras and quantum information is that\u2016<\/mml:mo>\u03a6<\/mml:mi>\u2297<\/mml:mo>id<\/mml:mtext><\/mml:mrow>k<\/mml:mi><\/mml:mrow><\/mml:msub>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub><\/mml:math>can grow withk<\/mml:mi><\/mml:math>. In general, the rate of growth is bounded by\u2016<\/mml:mo>\u03a6<\/mml:mi>\u2297<\/mml:mo>id<\/mml:mtext><\/mml:mrow>k<\/mml:mi><\/mml:mrow><\/mml:msub>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub>\u2264<\/mml:mo>k<\/mml:mi>\u2016<\/mml:mo>\u03a6<\/mml:mi>\u2016<\/mml:mo>1<\/mml:mn><\/mml:msub><\/mml:math>, and matrix transposition is the canonical example of a map achieving this bound. We prove that, up to an equivalence, the transpose is the unique map achieving this bound. The equivalence is given in terms of complete trace-norm isometries, and the proof relies on a particular characterization of complete trace-norm isometries regarding preservation of certain multiplication relations.We use this result to characterize the set of single-shot quantum channel discrimination games satisfying a norm relation that, operationally, implies that the game can be won with certainty using entanglement, but is hard to win without entanglement. Specifically, we show that the well-known example of such a game, involving the Werner-Holevo channels, is essentially the unique game satisfying this norm relation. This constitutes a step towards a characterization of single-shot quantum channel discrimination games with maximal gap between optimal performance of entangled and unentangled strategies.<\/jats:p>","DOI":"10.22331\/q-2018-02-05-51","type":"journal-article","created":{"date-parts":[[2018,2,5]],"date-time":"2018-02-05T17:56:59Z","timestamp":1517853419000},"page":"51","source":"Crossref","is-referenced-by-count":1,"title":["Characterization of linear maps onM<\/mml:mi>n<\/mml:mi><\/mml:msub><\/mml:math>whose multiplicity maps have maximal norm, with an application in quantum information"],"prefix":"10.22331","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3815-2072","authenticated-orcid":false,"given":"Daniel","family":"Puzzuoli","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Institute for Quantum Computing"},{"name":"University of Waterloo, Waterloo, Ontario, Canada"}]}],"member":"9598","published-online":{"date-parts":[[2018,2,5]]},"reference":[{"key":"1","doi-asserted-by":"publisher","unstructured":"R. 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