Hyperentangled Bell-states analysis of two-photon 2n-degree-of-freedom system

It is known that it is impossible to unambiguously distinguish all the Bell states in one system using only linear optics even if you also use hyperentanglement. However, we can find out how many Bell states can be unambiguously distinguished with a given n in two-photon 2n-degree-of-freedom hyperentangled system. This work, based on the criterion of Peter van Loock and Norbert Lutkenhaus, uses the symmetric matrix to express Bell states and the number of distinguished Bell states can be obtained by the rank of a matrix made with all the Bell states. The results show that with the increase of n, the number of distinguished Bell states increases exponentially by 2ⁿ⁺¹-1, while the efficiency of discrimination decreases exponentially by 1/2^n-1. This work lays a foundation for Bell-state analysis of three-photon system.