This paper investigates the degradation mechanism of the photon-number-encoded entanglement swapping protocol under the amplitude damping noise channel. By establishing a beam splitter physical model to simulate the energy dissipation process, the evolution density matrix of the input states Formula: see text and Formula: see text under independent noise channels is analytically derived, and the density matrix, fidelity, and concurrence of the target particle pair after entanglement swapping are presented. Furthermore, for the case where the initial states are maximally entangled states, this paper numerically simulates the variation curves of the fidelity and concurrence of the system after entanglement swapping with the noise parameter. The results show that as the noise intensity increases, both the fidelity and concurrence of the target system exhibit a decreasing trend. Simultaneously, due to the presence of noise, even if the input states are maximally entangled, the entanglement of the target system after swapping may be destroyed. Based on this, the study further deduces the constraint conditions required to maintain system entanglement in this scenario. Given the restrictive effect of these constraints on entanglement maintenance, this paper also specifically examines the fidelity and concurrence of the target state output from entanglement swapping in the case of 50 : 50 beam splitters when the input states are both maximally entangled. The research finds that in this case, since the constraints are not satisfied, even if the input states are maximally entangled, the entanglement of the output state completely disappears.
Yu et al. (Mon,) studied this question.