Wave-particle duality in single-photon entanglement

Single-photon entanglement, the entanglement between vacuum state and single-photon state, is essentially the entanglement of particle characteristics. Wave-particle duality indicates that wave property and particle property could coexist. Can we find evidence of wave-behavior entanglement in single-photon entanglement? Here, based on the construction of Bell-like inequality in wave space, we propose a method to verify the wave-particle duality in single-photon entanglement. First, we define the wave state in two-dimensional space relative to the photon number state, and propose a method to measure the wave state. The strong violation of Bell's inequality based on the joint measurement of wave states indicates the existence of single-photon entanglement, and the wave is the conjugate variable of the particle. Wave state entanglement obtained from Fourier transform of single photon entanglement and the corresponding measurement protocols will provide us with more information-carrying schemes in the field of quantum information. The difference in the representation in photon-number space and wave space implies the wave-particle duality of single photon entanglement.