Quantum physics phenomena,
entanglement and coherence,
are crucial
for quantum information protocols, but understanding these in systems
with more than two parts is challenging due to increasing complexity.
The W state, a multipartite entangled state, is notable for its robustness
and benefits in quantum communication. Here, we generate eight-mode
on-demand single-photon W states, using nanowire quantum dots and
a silicon nitride photonic chip. We demonstrate a reliable and scalable
technique for reconstructing the W state in photonic circuits using
Fourier and real-space imaging, supported by the Gerchberg-Saxton
phase retrieval algorithm. Additionally, we utilize an entanglement
witness to distinguish between mixed and entangled states, thereby
affirming the entangled nature of our generated state. The study provides
a new imaging approach of assessing multipartite entanglement in W
states, paving the way for further progress in image processing and
Fourier-space analysis techniques for complex quantum systems.