Abstract The teaching/learning of quantum mechanics via two-state systems is continuously spreading in secondary schools due to its promising results in physics education research. One possibility is the use of polarisation states of photons. This paper reports on a polarisation-based introduction to quantum mechanics in physics teacher education. A widely used school material prepares teacher trainees for their future work and also improves their conceptual knowledge. This part includes statistical calculations using only secondary school mathematics and a new formulation of the uncertainty relation, using only real numbers. The second step is to prepare the formalism of quantum mechanics using real two-dimensional vectors and matrices. Considering that students may not learn complex linear algebra, we offer a new way to introduce the complete formalism of two-state systems via circular polarisation providing a step-by-step exploration of complex quantum states. This points out the advantage of using complex linear algebra via a physical example, providing the opportunity to reach the elements of advanced quantum physics and quantum computing while deepening the physics background of the secondary school material.