Method of constructing the entangled state of the particle system

The entanglements on various dual properties, such as the wave–particle duality, spin-up and spin-down, and so on, have attracted enormous attention, but how to obtain their entanglement states easily was not clear until now for the particle system. Herein, a method is found for constructing entanglement states on various dual properties of the N -particle system by using the basis vectors of the wave function before the entanglement for the N particles, where the N particles are independent of each other before forming the N -particle system. Moreover, the wave function on the dual property of any one of the mutually independent N particles can be written as a linear combination of the same basis vectors before forming the N -particle system. Notably, the wave–particle entanglement state of two photons in the previous experimental work (Rab et al, Nat. Commun. 8 , 915 (2017)) and the entanglement state on the spin of two electrons have been obtained using the method.