Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Positive Maps and Entanglement in Real Hilbert Spaces

Positive Maps and Entanglement in Real Hilbert Spaces 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 less 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Henri Poincaré Springer Journals

Positive Maps and Entanglement in Real Hilbert Spaces

Loading next page...
 
/lp/springer-journals/positive-maps-and-entanglement-in-real-hilbert-spaces-EjgakNRdeN

References (81)

Publisher
Springer Journals
Copyright
Copyright © Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
1424-0637
eISSN
1424-0661
DOI
10.1007/s00023-023-01325-x
Publisher site
See Article on Publisher Site

Abstract

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 less 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.

Journal

Annales Henri PoincaréSpringer Journals

Published: Dec 1, 2023

There are no references for this article.