Remote Preparation of Entangled Photons

• The paper demonstrates a remote state preparation (RSP) protocol where LOV prism pairs generate spatially structured spin–orbit entangled OAM lattices.
• It details an experimental setup using a type-II ppKTP source in a Sagnac interferometer, achieving high polarization Bell-state fidelity and pixel-wise state tomography.
• The work highlights applications in high-dimensional quantum communication, quantum sensing, and scalable quantum networks through deterministic photonic state engineering.

Remote preparation of entangled photons refers to protocols in which the measurement or manipulation of a subsystem at one location deterministically or heraldedly prepares entangled photon states at distant locations. In contrast to quantum teleportation—where the goal is to faithfully transmit an unknown quantum state via pre-shared entanglement and classical communication—remote state preparation (RSP) leverages prior knowledge of the target state at the sender to simplify the local operations, classical communication, and quantum resource requirements. RSP protocols are central to distributed quantum information processing, quantum communication networks, high-dimensional quantum encoding, and quantum state engineering in the laboratory.

1. Theoretical Framework: Spin–Orbit and OAM Lattice States

The prototypical instance of remote preparation of entangled photons in spatial degrees of freedom is realized using polarization–orbital angular momentum (OAM) entangled pairs. Consider the polarization Bell state in the circular basis:

$$|\Phi^+\rangle = \frac{1}{\sqrt{2}} \left( |R\rangle_1 |L\rangle_2 + |L\rangle_1 |R\rangle_2 \right)$$

with $|R\rangle = (|H\rangle + i|V\rangle)/\sqrt{2}$, $|L\rangle = (|H\rangle - i|V\rangle)/\sqrt{2}$.

To engineer spatially structured quantum states, the signal photon is passed through a series of lattice-of-optical-vortices (LOV) prism pairs, each inducing polarization-dependent transverse phase gradients:

$$U_x = \exp\left[ i \frac{\pi}{a}(x-x_0)\sigma_x \right], \quad U_y = \exp\left[ i \frac{\pi}{a}(y-y_0)\sigma_z \right]$$

where σ_x, σ_z are Pauli matrices, a is the lattice period, and (x_0, y_0) specify the lattice origin.

After N=2 such prism pairs, the two-photon state is:

$$|\Psi_{\mathrm{LOV}}^{N=2}\rangle(x, y) = \alpha(x,y) \left[ (U_x U_y)^2 \otimes I \right] |\Phi^+\rangle$$

where α(x, y) is a Gaussian envelope. The output is a spin–orbit entangled state, with spatial functions A(x,y), B(x,y) modulating the local mixture of |L⟩ and |R⟩. Measuring one photon's polarization remotely prepares the partner photon in a spatially structured OAM lattice pattern determined by the choice of analyzer setting (Cameron et al., 2021).

2. Experimental Realization with LOV Prism Pairs

The experimental apparatus for remote preparation of spin–orbit entangled OAM lattices is built on several key elements:

• Entangled source: A Sagnac interferometer containing a type-II periodically poled KTP crystal (ppKTP, 10 mm) produces 808 nm polarization-entangled photons with ~96% Bell-state fidelity. The pump is a 404 nm CW diode laser.
• LOV prism pairs: Each pair consists of birefringent wedges with fast axes at 45° with respect to each other. One wedge imposes a polarization-dependent phase gradient along x (∝xσ_x), the orthogonal wedge along y (∝yσ_z). Two pairs generate a 2D periodic OAM lattice.
• Beam conditioning: A telescope magnifies the signal path by 8.3× prior to the LOVs, setting a physical lattice constant (~0.52 mm at the imaging plane) and maximizing the number of accessible lattice sites on the detection camera.

After the signal photon passes through the LOV prism pairs, its polarization and OAM are entangled in a periodic structure. The idler photon's polarization is analyzed in a tomographically complete basis (H, V, D, R), triggering a 3 ns gate on an electron-multiplying intensified CCD (emICCD) camera for spatially resolved single-photon imaging.

3. RSP Protocol and Tomographic Verification

The RSP sequence unfolds as follows:

1. Entangled resource generation: A polarization Bell pair is produced via type-II SPDC in the Sagnac-interferometer.
2. LOV-induced spin–orbit coupling: The magnified signal photon traverses two LOV pairs, imprinting OAM-lattice structure conditionally on its polarization.
3. Idler projective measurement: The idler's polarization is projectively measured, and an APD click triggers the emICCD gate for subsequent coincidence detection.
4. Remote preparation and imaging: The post-selected measurement outcome determines which spin–orbit structured state the signal photon collapses onto. No active unitary correction is required on the signal photon—the idler measurement and feedforward suffice.
5. Polarization tomography and imaging: Half-wave and quarter-wave plates plus a PBS analyze the signal photon polarization. Imaging onto the camera records the spatial intensity distribution.

Full two-qubit tomography is accomplished for each spatial pixel. For every idler–signal polarization basis combination (4×4=16), 2000 camera exposures are performed at 15 kHz. Standard maximum-likelihood algorithms reconstruct the 4×4 local density matrix ρ(x, y) at each pixel.

4. Metrics, Entanglement Witnesses, and Limitations

For each spatial mode (pixel), the fidelity with the four Bell states is computed:

$$F^k(x, y) = \text{Tr}[\rho(x, y) |B_k\rangle\langle B_k|], \quad B_k \in \{\Phi^+,\Phi^-,\Psi^+,\Psi^-\}$$

Any pixel where $F > 0.5$ constitutes a certified entanglement witness, as separable states cannot exceed this threshold. In theory, 85.7% of pixels should exceed this limit, but the experiment observes 42.5%, due to alignment errors, background counts, finite camera quantum efficiency, imperfect SPDC source fidelity, and modal mismatch in imaging.

Principal error sources:

• LOV prism misalignment: Imperfect calibration induces spatial phase errors that degrade the lattice pattern's fidelity.
• Background and detector noise: Reduces signal-to-noise, particularly in higher-order spatial modes.
• Quantum efficiency: The finite efficiency of the emICCD and APDs limits data rates and detection probability.
• Mode mismatch: Non-ideal overlap between LOV output and detection optics reduces the projective purity.

5. Applications and Broader Implications

Remote preparation of entangled photons in spatially structured, multi-DOF states enables several advanced applications:

• Quantum sensing of periodic structures: The spin–orbit OAM lattices sensitively probe spatial phase features, suitable for metrological applications involving periodicity and spatial inhomogeneity.
• High-dimensional quantum communication: The OAM lattice provides access to an expanded Hilbert space, while the polarization degree of freedom supports efficient state characterization and manipulation.
• Quantum control in synthetic lattices: RSP protocols may be extended to superpose OAM lattices with different periodicities, topological charges, or spatial symmetries, relevant to simulation of synthetic lattice systems.
• Matter-wave analogues: Formal parallels between photonic spin–orbit lattices and neutron spin–orbit states suggest that similar protocols could remotely prepare spatially structured states in matter-wave systems, relevant for quantum materials characterization.

6. Summary Table: Key Experimental Parameters and Metrics

Aspect	Value/Description	Note
SPDC Source	404 nm CW pump, type-II ppKTP	Sagnac interferometer geometry
Entanglement Fidelity	~96%	Polarization Bell state
LOV Lattice Constant	~0.52 mm at camera	Set by Δn, θ, λ, and telescopes
Camera	PI-MAX4 1024 EMB, emICCD	3 ns gate; pixel-wise tomography
Tomography Combinations	16 per pixel (4 idler × 4 signal)	H, V, D, R bases
Bell Witness Fraction	42.5% pixels with F > 0.5	Theoretical: 85.7%
Principal Infidelity	LOV prism alignment, background counts	Limits spatially resolved entanglement

7. Outlook and Future Directions

The demonstration of pixel-wise RSP for spin–orbit entangled photon lattices pushes the frontier of spatially structured, multi-DOF quantum optics. It establishes a platform for:

• Scalable high-dimensional photonic state preparation, tomography, and characterization with minimal feedforward requirements.
• Quantum sensing protocols where spatial entanglement and OAM lattice structure enhance sensitivity to periodic or topological sample features.
• Integration with advanced detection (e.g., camera arrays, single-photon megapixel sensors) for improved ensemble statistics and higher-dimensional quantum state engineering.
• Transfer and adaptation of RSP protocols to other systems, such as synthetic matter-wave lattices or electron vortex beams.

No active correction is required in the presented protocol—heralded state selection via projective measurement and feedforward suffices for deterministic RSP of targeted spin–orbit lattice states in the signal arm. This operational simplicity, combined with spatial multiplexing capacity, underpins deployment potential in quantum networks, quantum metrology devices, and programmable quantum simulators (Cameron et al., 2021).