Positive maps and entanglement in real Hilbert spaces

TL;DR

This paper explores positive maps on real Hilbert spaces, revealing fundamental differences from complex spaces and their implications for quantum entanglement, including conditions for positive complexification and properties of entanglement-breaking maps.

Contribution

It provides a necessary and sufficient condition for real maps to admit positive complexification and connects entanglement properties in real versus complex quantum mechanics.

Findings

Identified fundamental differences between real and complex positive maps.

Established conditions for positive complexification of real maps.

Disproved a real version of the PPT-squared conjecture in dimension 2.

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 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.