A dual-rail photonic qubit gate needs its two photons to be quantum-mechanically identical in every degree of freedom except the one carrying information. The coincidence probability follows:
P(Δt) = ½·[1 − V·exp(−(Δt/τ)²)]
- Δt — timing mismatch between the two arms (path-length / time-bin error)
- τ — photon coherence time (set by source bandwidth)
- V — visibility: 1 means perfectly indistinguishable, 0 means fully distinguishable (e.g. crossed polarizations)
At Δt=0 with V=1 the dip bottoms out at P=0 — perfect bunching, zero coincidences. This is exactly the interference measurement used to certify single-photon sources (SPDC, quantum dots) before wiring them into a linear-optical quantum circuit.