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Article: Counterexamples to the extendibility of positive unital norm-one maps
Title | Counterexamples to the extendibility of positive unital norm-one maps |
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Authors | |
Keywords | Operator system Positive linear maps Quantum channel |
Issue Date | 10-Jan-2023 |
Publisher | Elsevier |
Citation | Linear Algebra and its Applications, 2023, v. 663, p. 102-115 How to Cite? |
Abstract | Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C* -algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space. |
Persistent Identifier | http://hdl.handle.net/10722/331902 |
ISSN | 2023 Impact Factor: 1.0 2023 SCImago Journal Rankings: 0.837 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Chiribella, Giulio | - |
dc.contributor.author | Davidson, Kenneth R | - |
dc.contributor.author | Paulsen, Vern I | - |
dc.contributor.author | Rahaman, Mizanur | - |
dc.date.accessioned | 2023-09-28T04:59:29Z | - |
dc.date.available | 2023-09-28T04:59:29Z | - |
dc.date.issued | 2023-01-10 | - |
dc.identifier.citation | Linear Algebra and its Applications, 2023, v. 663, p. 102-115 | - |
dc.identifier.issn | 0024-3795 | - |
dc.identifier.uri | http://hdl.handle.net/10722/331902 | - |
dc.description.abstract | <p>Arveson's extension theorem guarantees that every completely positive map defined on an operator system can be extended to a completely positive map defined on the whole C* -algebra containing it. An analogous statement where complete positivity is replaced by positivity is known to be false. A natural question is whether extendibility could still hold for positive maps satisfying stronger conditions, such as being unital and norm 1. Here we provide three counterexamples showing that positive norm-one unital maps defined on an operator subsystem of a matrix algebra cannot be extended to a positive map on the full matrix algebra. The first counterexample is an unextendible positive unital map with unit norm, the second counterexample is an unextendible positive unital isometry on a real operator space, and the third counterexample is an unextendible positive unital isometry on a complex operator space.<br></p> | - |
dc.language | eng | - |
dc.publisher | Elsevier | - |
dc.relation.ispartof | Linear Algebra and its Applications | - |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.subject | Operator system | - |
dc.subject | Positive linear maps | - |
dc.subject | Quantum channel | - |
dc.title | Counterexamples to the extendibility of positive unital norm-one maps | - |
dc.type | Article | - |
dc.identifier.doi | 10.1016/j.laa.2023.01.003 | - |
dc.identifier.scopus | eid_2-s2.0-85146579713 | - |
dc.identifier.volume | 663 | - |
dc.identifier.spage | 102 | - |
dc.identifier.epage | 115 | - |
dc.identifier.eissn | 1873-1856 | - |
dc.identifier.isi | WOS:000964268600001 | - |
dc.identifier.issnl | 0024-3795 | - |