LOQC State Generation

Step-by-Step LOQC State Generation

1. Photon Source Preparation
   Begin with spontaneous parametric down-conversion (SPDC) to generate entangled photon pairs:
   \(|\psi\rangle = \frac{1}{\sqrt{2}}(|H\rangle_1|H\rangle_2 + |V\rangle_1|V\rangle_2)\)

2. Polarization Encoding
   Use wave plates to encode qubits in photon polarization states:
   \(|0\rangle \equiv |H\rangle, \quad |1\rangle \equiv |V\rangle\)

3. Beam Splitter Operation
   Apply a 50:50 beam splitter transformation:
   \(\hat{a}^\dagger \rightarrow \frac{1}{\sqrt{2}}(\hat{b}^\dagger + \hat{c}^\dagger)\)
   \(\hat{d}^\dagger \rightarrow \frac{1}{\sqrt{2}}(\hat{b}^\dagger - \hat{c}^\dagger)\)

4. Post-Selection
   Detect photons in specific output modes to herald successful state creation (e.g., Bell state generation requires coincidence detection)

5. Feed-Forward Correction
   Apply conditional phase shifts based on measurement outcomes:
   \(U_{ff} = e^{i\pi|1\rangle\langle 1|}\)

6. Verification
   Perform quantum state tomography to confirm the generated state’s fidelity

Key optical components required:

• Nonlinear crystals (BBO/PPKTP)
• Polarizing beam splitters
• Half-wave and quarter-wave plates
• Single-photon detectors

The entire process can be represented as a quantum circuit:
\(\text{SPDC} \rightarrow \text{BS} \rightarrow \text{PS} \rightarrow \text{PBS} \rightarrow \text{Det.}\)

Where BS = Beam Splitter, PS = Phase Shifter, PBS = Polarizing Beam Splitter, Det. = Detection. The exact sequence depends on the target state (e.g., GHZ, cluster, or graph states).