Heralded qudit-based high-dimensional entanglement generation for hybrid photon-emitter system by waveguide-mediated scattering

Quantum entanglement systems based on qudits dilate high-dimensional (HD) state spaces and enhance resistance to loss in quantum information processing (QIP). To fully exploit this potential, effective schemes for generating HD entanglement are crucial. In this paper, we propose a flexible heralded scheme generating random four-dimensional (4D) two-qudit maximal entanglement for a hybrid photon-emitter system by entering different input ports. This approach can be further extended to prepare 4D $n$-qudit $(n\ensuremath{\ge}3)$ maximal entanglement utilizing the 4D single-qudit ${Z}^{m}(m=1,2,3)$ gate for the first qudit and ${X}^{m}$ gate for the other qudits (except the second qudit). For the hybrid system, the first 4D qudit is encoded on the hybrid polarization-path states of a flying photon, while the second and subsequent 4D qudits are represented by two stationary emitters coupled to the respective 1D waveguide. The qudit-encoded hybrid HD entanglement offers advantages over economizing quantum resource without any auxiliary qudits and obtaining robust fidelities of various HD entanglement by the error-detected mechanism of the emitter-waveguide systems. Moreover, the proposed protocol can be spread to generate $d\mathrm{D}$ $n$-qudit $(d\ensuremath{\ge}{2}^{p+1},$ $n,p=2,3,\ensuremath{\cdots})$ entangled states, further broadening its applicability in HD QIP.