Schemes for fusing photonic W-state simultaneously without qubit loss via weak cross-Kerr nonlinearities

We propose three schemes to generate polarization W-states of size $$n+m$$ , $$n+m+t$$ , and $$n+m+t+z$$ without qubit loss, i.e., large-sized $$W_{n+m}$$ ( $$W_{n+m+t}$$ and $$W_{n+m+t+z}$$ ) is generated from one $$W_n$$ and one $$W_m$$ (one $$W_n$$ , one $$W_m$$ , and one $$W_t$$ , and one $$W_n$$ , one $$W_m$$ , one $$W_t$$ , and one $$W_z$$ ). The qubit-loss-free fusion mechanism increases the size of the fusion W states essentially, i.e., reduces the fusion steps, which makes the implementations of the schemes possible within current technology. Ancillary photons, controlled-NOT, Toffoli, Fredkin, and partial-SWAP gates are not required in our schemes. In addition, the number of the coherence states is reduced from 3 to 2, and path couplers are not employed. There is no complete failure output, and all garbage states are recyclable in the $$W_{n+m}$$ fusion scheme.