Counterexamples to the extendibility of positive unital norm-one maps
| Autor: | Giulio Chiribella, Kenneth R. Davidson, Vern I. Paulsen, Mizanur Rahaman |
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| Přispěvatelé: | The University of Hong Kong (HKU), University of Waterloo [Waterloo], Traitement optimal de l'information avec des dispositifs quantiques (QINFO), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Université Grenoble Alpes (UGA)-Inria Lyon, Institut National de Recherche en Informatique et en Automatique (Inria), HEP, INSPIRE, Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2023 |
| Předmět: |
Quantum Physics
Numerical Analysis Algebra and Number Theory [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Mathematics - Operator Algebras FOS: Physical sciences [PHYS.MPHY] Physics [physics]/Mathematical Physics [math-ph] algebra Functional Analysis (math.FA) Mathematics - Functional Analysis [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph] isometry FOS: Mathematics Discrete Mathematics and Combinatorics Geometry and Topology Operator Algebras (math.OA) Quantum Physics (quant-ph) [PHYS.QPHY] Physics [physics]/Quantum Physics [quant-ph] |
| Zdroj: | Linear Algebra Appl. Linear Algebra Appl., 2023, 663, pp.102-115. ⟨10.1016/j.laa.2023.01.003⟩ |
| Popis: | 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. Comments are welcome |
| Databáze: | OpenAIRE |
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