The electron-positron annihilation into two photons is a standard technology in medicine to observe e.g. metabolic processes in human bodies. A new tomograph, the J-PET, will provide the possibility to observe not only direct positron-electron annihilations but also the 2-photon and 3-photons decay of positronium atoms. Moreover, the polarisation properties of these photons may become feasible over Compton scattering processes. This talk discusses the theoretically predicted entanglement of the two- and the three-photon states and outlines how it can be detected. In particular the three-photon state exhibits an interesting entanglement, namely it is genuinely multipartite entangled, a type of entanglement involving all degrees of freedom. Surprisingly, even if all spin eigenstates mix, a situation expected in the human body, still the entanglement due to symmetrization symmetries survives. Once this bipartite or/and multipartite entanglement can be experimentally observed novel biological indicators, e.g. relating cancer detection and entanglement in the positronium decay, may become a standard technology for doctors.
 B.C. Hiesmayr and P. Moskal , Witnessing Entanglement In Compton Scattering Processes Via Mutually Unbiased Bases, arXiv:1807.04934
 B.C. Hiesmayr and P. Moskal , Genuine Multipartite Entanglement in the 3-Photon Decay of Positronium, Scientific Reports 7: 15349 (2017).
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