Genuine Multipartite Entanglement in the 3-Photon Decay of Positronium

Abstract

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 will provide the possibility to observe not only direct e ⁺ e ^- annihilations but also the 3 photons from the decay of ortho-positronium atoms formed in the body. We show in this contribution that the three-photon state with respect to polarisation degrees of freedom depends on the angles between the photons and exhibits various specific entanglement features. In particular genuine multipartite entanglement, a type of entanglement involving all degrees of freedom, is subsistent if the positronium was in a definite spin eigenstate. Remarkably, when all spin eigenstates are mixed equally, entanglement -and even stronger genuine multipartite entanglement- survives. Due to a "symmetrization" process, however, Dicke-type or W-type entanglement remains whereas GHZ-type entanglement vanishes. The survival of particular entanglement properties in the mixing scenario may make it possible to extract quantum information in the form of distinct entanglement features, e.g., from metabolic processes in human bodies.

Figure 1

This graphic shows schematically how from an isotope typically used in standard PET-therapy, e.g. FDG-18 (fludeoxyglucose), positronium is generated that decays into three photons with wave vectors which lie in one plane due to energy and momentum conservation.

Figure 2

These contour plots show the maximum taken over all single photon energies of three photons for the allowed angles ${\tilde{\mathrm{\Theta }}}_{ab}$ (x-axis) and ${\tilde{\mathrm{\Theta }}}_{bc}$ (y-axis). Three kinematically different regions emerge. A forbidden region (III) where the momentum conservation does not hold since all wave vectors point into one half of the plane. Another forbidden region (II) where one photon has zero energy. And a physically relevant region (I), here the energies are not extremal. This plot agrees with Figure 8 of ref., where also a Dalitz plot is shown for this case.

Figure 3

These contour plots show the function (a) Q _SEP, (b) Q _GHZ and (c) Q _W for the pure state $|{\psi }_{pure}({\tilde{\mathrm{\Theta }}}_{ab},{\tilde{\mathrm{\Theta }}}_{bc})\rangle$ for each ${\tilde{\mathrm{\Theta }}}_{ab}$ (x-axis) and ${\tilde{\mathrm{\Theta }}}_{bc}$ (y-axis) (optimized via local unitaries). Q _SEP is always greater than zero, indeed even $\ge \frac{1}{2}$, thus proving entanglement for all possible decay scenarios. The quantities detecting genuine multipartite entanglement Q _W, Q _GHZ are greater than zero, thus detecting genuine multipartite entanglement, however, their values differ.

Figure 4

This contour plot shows the tangle minus the two concurrences, τ _i|jk − C(ρ _ij)² − C(ρ _ij)², which is equal for any permutation of the three photons.

Figure 5

These three contour plots show (a) Q _SEP, (b) Q _GHZ and (c) Q _W for the state mixed equally between all three possible quantum states $s_{\hat{\overrightarrow{n}}}=\mathrm{0,}+1,-1$, equation 17. Still genuine multipartite entanglement is revealed for some scenarios $({\tilde{\mathrm{\Theta }}}_{ab},{\tilde{\mathrm{\Theta }}}_{bc})$. The criterion Q _W detecting W-type of genuine multipartite entanglement is by far more sensitive to reveal genuine multipartite entanglement.