Fig. 1. (a) Schematic of quantum state manipulation with a parallel BS metasurface. (b) Parallel BS processes on a single metasurface. (c) The model of linked beam splitters for parallel BSs.

Fig. 2. Schematic of two-photon entangled state preparation with parallel BS of metasurface. When two LCP photons along paths 0 and +1 are incident to the metasurface, the entangled state 12(|L⟩0|L⟩+1−|R⟩+1|R⟩+2) is prepared after a postselection. Here, SPD is an abbreviation of a single photon detector.

Fig. 3. Schematic of four kinds of two-photon entangled state preparation with parallel BS of gradient metasurface. (a) |ψin⟩=|R⟩0|R⟩+1 and |ψout⟩post=(|R⟩0|R⟩+1−|L⟩0|L⟩−1)/2. (b) |ψin⟩=|L⟩0|R⟩+1 and |ψout⟩post=(|LL⟩0+|RR⟩+1)/2. (c) |ψin⟩=|L⟩−1|R⟩+1 and |ψout⟩post=(|L⟩0|L⟩−1+|R⟩0|R⟩+1)/2. (d) |ψin⟩=|H⟩−1|H⟩+1 and |ψout⟩post=(|L⟩0|L⟩−2+|R⟩0|R⟩+2)/2. The splitting parameters are Δ=π/4 for panels (a)–(c) and Δ=π/2 for panel (d). The postselection with a success probability of 50% is done by single-photon detection in path 0 for panels (a), (c), and (d), whereas for panel (b), the postselection is not required. Here, |ψout⟩post has been normalized, with the global phase factor being omitted.

Fig. 4. Schematic of three-photon entangled state preparations with parallel BS of gradient metasurface. (a) Three-photon entangled state preparation with input state |ψin⟩=|R⟩−1|R⟩0|R⟩+1, which needs single-photon detection coincidence in paths 0 and −1 for postselection. (b) Three-photon entangled state preparation with input state |ψin⟩=|H⟩−2|H⟩0|H⟩+2, which needs single-photon detection coincidence in paths +1 and −1 for postselection. The splitting parameters for panels (a) and (b) are Δ=π/4 and Δ=π/2, respectively. (c) Extended scheme of Figs. 3(a) or 3(d) for the preparation of an N-photon entangled state, which needs (N−1)-fold single-photon detection coincidence for postselection.

Fig. 5. Schematic of entanglement fusion and conversion with parallel BS of a metasurface. (a) Fusion and conversion of two pairs of Bell states. (b) Fusion and conversion of three pairs of Bell states. Here, the BS parameter of the metasurface is Δ=π/2.