The sliders set a single-photon qubit state |ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩, drawn as an arrow on the Bloch sphere. Different quantum hardware wants different photon colours — a solid-state memory or superconducting-qubit link is efficient near the input wavelength, long-haul fibre is efficient only in the telecom bands. Both converters below change the photon's wavelength; only one preserves what it was carrying.
Naive: |ψ⟩ --[detect]--> classical bit --[re-emit]--> |0⟩ or |1⟩ at λ_out
Coherent: |ψ⟩ + pump(λ_p) --[χ⁽²⁾ crystal]--> |ψ⟩ at λ_out (state untouched)
- Naive re-emission — absorbing the photon to measure "which colour to re-emit" is itself a projective measurement. It collapses the qubit onto |0⟩ or |1⟩ with Born-rule probabilities cos²(θ/2) / sin²(θ/2), then a classical source re-emits a definite bit at the new wavelength. Coherence (the off-diagonal |ρ₀₁|) drops to zero every time — only classical basis states (θ=0° or 180°) survive by luck.
- Quantum-coherent conversion — a strong pump laser mixes with the photon inside a nonlinear (χ⁽²⁾) crystal via sum- or difference-frequency generation. Energy conservation shifts the photon's frequency, but the process never measures which state it was in, so the full superposition — amplitude and phase — transfers to the new colour. Fidelity stays ≈100% (small noise here just represents non-unit conversion efficiency).
- Fidelity F = |⟨ψin|ψout⟩|² compares the output state to the input state directly. Coherence is the magnitude of the off-diagonal density-matrix term, i.e. how much superposition survives.
- In a real lab this is verified the same way: send one photon of an entangled pair through the converter and check the Bell-inequality violation (or Bloch-sphere interference) still holds with its untouched partner — that is exactly what a fidelity ≈1 readout here stands in for.