Considering two-dimensional electron gases under a perpendicular magnetic field, we pinpoint a specific kind of long-range bipartite entanglement of the electronic motions. This entanglement is achieved through the introduction of bicomplex spinorial eigenfunctions admitting a polar decomposition in terms of a real modulus and three real phases. Within this bicomplex geometry the cyclotron motions of two electrons are intrinsically tied, so that the highlighted eigenstates of the kinetic energy operator actually describe the free motion of a genuine electron pair. Most remarkably, these states embody phase singularities in the four-dimensional (4D) space, with singular points corresponding to the simultaneous undetermination of the three phases. Because the entanglement between the two electrons forming a pair, as well as the winding and parity quantum numbers characterizing the 4D phase singularity, are topological in nature, we expect them to manifest some robustness in the presence of a smooth disorder potential and an electron-electron interaction potential. The relevance of this effective approach in terms of 4D vortices of electron pairs is discussed in the context of the fractional quantum Hall effect.
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