States and channels in quantum mechanics without complex numbers

TL;DR

This paper explores a real-number based formulation of quantum mechanics, simplifying state representation and reducing computational complexity, but at the cost of losing hermiticity in quantum states.

Contribution

It introduces a real-number framework for quantum states and channels, providing a computationally efficient alternative to standard complex-number quantum mechanics.

Findings

Real-only quantum states require n^2 real numbers for n-dimensional systems.

The approach simplifies quantum state manipulation and simulation.

The representation lacks hermiticity, affecting certain quantum properties.

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.