Positive Maps and Entanglement in Real Hilbert Spaces
| Autor: | Giulio Chiribella, Kenneth R. Davidson, Vern I. Paulsen, Mizanur Rahaman |
|---|---|
| Přispěvatelé: | Rahaman, Mizanur, City University of Hong Kong [Hong Kong] (CUHK), Department of Pure Mathematics [Waterloo], 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) |
| Rok vydání: | 2023 |
| Předmět: |
[PHYS]Physics [physics]
Quantum Physics Nuclear and High Energy Physics Mathematics - Operator Algebras FOS: Physical sciences Statistical and Nonlinear Physics [MATH] Mathematics [math] [PHYS] Physics [physics] Functional Analysis (math.FA) Mathematics - Functional Analysis FOS: Mathematics [MATH]Mathematics [math] Quantum Physics (quant-ph) Operator Algebras (math.OA) Mathematical Physics |
| Zdroj: | Annales Henri Poincaré. |
| ISSN: | 1424-0661 1424-0637 |
| Popis: | The theory of positive maps plays a central role in operator algebras and functional analysis, and has countless applications in quantum information science. The theory was originally developed for operators acting on complex Hilbert spaces, and little is known about its variant on real Hilbert spaces. In this article we study positive maps acting on a full matrix algebra over the reals, pointing out a number of fundamental differences with the complex case and discussing their implications in quantum information. We provide a necessary and sufficient condition for a real map to admit a positive complexification, and connect the existence of positive maps with non-positive complexification with the existence of mixed states that are entangled in real Hilbert space quantum mechanics, but separable in the complex version, providing explicit examples both for the maps and for the states. Finally, we discuss entanglement breaking and PPT maps, and we show that a straightforward real version of the PPT-squared conjecture is false even in dimension 2. Nevertheless, we show that the original PPT-squared conjecture implies a different conjecture for real maps, in which the PPT property is replaced by a stronger property of invariance under partial transposition (IPT). When the IPT property is assumed, we prove an asymptotic version of the conjecture. Comment: Updated version. To appear in Annales Henri Poincar\'e |
| Databáze: | OpenAIRE |
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