Spatial electron-photon entanglement

Free electron beams and their quantum coupling with photons are attracting a rising interest due to the basic questions they address and the innovative technology these particles are involved in, such as microscopy, spectroscopy, and quantum computation. This work proposes spatially encoded coupling, where a dual-rail-like electron (e-) state is entangled to a dual-rail photonic qubit. Thus, it complements the leading current concept of the coupling, which links photonic Fock states with the e- energy. Importantly, we show that a spatial electron-photon pair can reach maximal entanglement and propose mutually unbiased bases for verifying the entanglement in a realistic experimental apparatus. Although the energy-level coupling of an e- beam with photonic excitations is well understood, we show that a naïve approach for extending such interaction from a single to a double path results in a retrocausal paradox. In the future, the atomic scale precision of electron microscopes can harness such position-encoded free-e- entangled qubits for novel quantum sensing and quantum transduction.