Photon-number entanglement from cascaded parametric down-conversion

We propose a protocol for generating photon-number entanglement through sequential parametric down-conversion processes. This protocol is fully compatible with linear optics experiments and can be easily implemented as an extension of the path-identity scheme. We present a detailed scheme for the creation of heralded two-mode photon-number high-dimensional entanglement with the form of $|00\ensuremath{\rangle}+|11\ensuremath{\rangle}+|22\ensuremath{\rangle}$. In addition, the protocol enables us to easily generate various photon-number-entangled states, which have potential applications in quantum communication, quantum precision measurements, and continuous-variable quantum information.