Effective schemes for fusion of hyperentangled W states

Hyperentangled states are fascinating resources in quantum information processing as they can significantly increase the channel capacity and enhance noise resistance. We explore a hyperfusion mechanism to fuse one n photon hyper-W state and one m-photon hyper-W state into a large-scale (n+m-2)-photon hyper-W state. Another mechanism to fuse one n-photon hyper-W state, one m-photon hyper-$W$ state, and one $t$-photon hyper-W state into an (n+m+t-3)-photon hyper-W state is also proposed. These two hyperfusion mechanisms are constructed employing only polarizing beam splitters, balanced beam splitters, half-wave plates, single-photon detectors, and cross-Kerr nonlinearities. Conditional quantum gates, path couplers, and ancillary photons are not required in our constructions. Moreover, our fused $W$ states are hyperentangled in the polarization and spatial degrees of freedom of single-photon systems. The presence of only one garbage output state demonstrates that high efficiency can be achieved in our schemes.