Positive maps and entanglement in real Hilbert spaces

Understand

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.