States and channels in quantum mechanics without complex numbers

Abstract: In the presented note we aim at exploring the possibility of abandoning complex numbers in the representation of quantum states and operations. We demonstrate a simplified version of quantum mechanics in which the states are represented using real numbers only. The main advantage of this approach is that the simulation of the $n$-dimensional quantum system requires $n^2$ real numbers, in contrast to the standard case where $n^4$ real numbers are required. The main disadvantage is the lack of hermicity in the representation of quantum states. Using Mathematica computer algebra system we develop a set of functions for manipulating real-only quantum states. With the help of this tool we study the properties of the introduced representation and the induced representation of quantum channels.