An experiment for the advanced undergraduate laboratory which allows students to study the effect of photon polarisation in Compton scattering and to explore quantum entanglement is described. The quantum entangled photons are produced through electron-positron annihilation in the S-state, and their polarisations are analysed using the Compton scattering cross-section dependence on the photon polarisation. The experiment was equipped with off-the-shelf detectors and electronic units. Finite geometry effects are discussed and investigated with the use of a Geant4based simulation.