Wavefront correction of high-dimensional two-photon states via coherence-entanglement transfer (2509.04170v1)

Abstract: Reliable transmission of quantum optical states through real-world environments is key for quantum communication and imaging. Yet, aberrations and scattering in the propagation path can scramble the transmitted signal and hinder its use. A typical strategy is to employ a classical beacon beam to learn and then correct for the wavefront distortions. However, relying on a separate light source increases the overhead in the experimental apparatus. Moreover, the beacon light must closely match the non-classical state in polarization, wavelength, and even temporal bandwidth, which is highly challenging in practice. Here, we introduce a fast and efficient wavefront correction approach where we use the quantum state itself to correct for optical distortion. Via pump shaping, we control the degree of entanglement in the spatially-entangled two-photon state so that it behaves either as a high-dimensional entangled state or as a classical coherent state. The latter case is used to efficiently measure the transmission matrix of the propagation channel and correct its distortions with a spatial light modulator, thereby enabling the transmission of the high-dimensional entangled state with minimal errors. Our approach paves the way for the practical implementation of quantum imaging and communication protocols based on high-dimensional spatially entangled states.

• The paper introduces a beacon-free protocol that exploits the duality between spatial coherence and entanglement for wavefront correction in high-dimensional two-photon states.
• The paper uses SPDC to generate entangled photon pairs and a spatial light modulator to measure the transmission matrix and restore quantum correlations in scattering media.
• The paper demonstrates that tuning the Schmidt number enhances wavefront optimization efficiency, offering improved performance for quantum communication and imaging applications.

Wavefront Correction of High-Dimensional Two-Photon States via Coherence-Entanglement Transfer

Introduction and Motivation

The transmission of high-dimensional spatially entangled quantum states through complex media is a central challenge in quantum communication and imaging. Aberrations and scattering in the propagation channel—such as atmospheric turbulence or biological tissue—degrade quantum correlations, limiting the practical deployment of quantum-enhanced protocols. Conventional wavefront correction strategies rely on a classical beacon beam to measure and compensate for distortions, but this approach introduces significant overhead and mode-matching constraints in polarization, wavelength, and temporal bandwidth. The paper presents a protocol that obviates the need for a classical beacon by exploiting the duality between spatial coherence and entanglement in two-photon states, enabling wavefront correction using the quantum state itself.

Experimental Protocol and Setup

The experimental protocol leverages Spontaneous Parametric Down-Conversion (SPDC) to generate spatially entangled photon pairs. By shaping the pump beam, the entanglement dimensionality (quantified by the Schmidt number $K$) and spatial coherence of the two-photon state can be tuned. A focused pump produces a low-$K$ state with high spatial coherence, mimicking classical light and yielding high-contrast intensity speckle patterns after propagation through a scattering medium. This enables efficient measurement of the transmission matrix (TM) using intensity feedback and a spatial light modulator (SLM). The same SLM is then used to correct the wavefront for the high-dimensional entangled state, restoring quantum correlations without the need for a separate classical reference.

Figure 1: Schematic of the experimental setup for wavefront correction using SPDC-generated two-photon states and a spatial light modulator.

Coherence-Entanglement Transfer and Transmission Matrix Measurement

The core concept is the transfer of entanglement to spatial coherence by adjusting the pump waist $w$. The two-photon wavefunction is modeled as a double-Gaussian distribution, with the first-order spatial coherence function $g^{(1)}$ and Schmidt number $K$ given by:

$$g^{(1)}(\vec{r},-\vec{r}) = \exp\left(-\frac{|\vec{r}|^2}{8w^2}\frac{(24\pi w^2 -L\lambda_p)^2}{(24\pi w^2 +L\lambda_p)L\lambda_p}\right)$$

$$K = \frac{1}{4} \left( \frac{24\pi w^2 +L\lambda_p}{\sqrt{24\pi w^2}\sqrt{L\lambda_p}} \right)^2$$

As $w$ decreases, spatial coherence increases and entanglement dimensionality decreases, facilitating TM measurement via intensity speckle patterns. The TM is measured with a $1024 \times 14641$ matrix using an EMCCD camera and SLM, enabling phase pattern generation for wavefront correction.

Figure 2: Intensity images and speckle patterns for low-Schmidt-number two-photon states before and after wavefront correction.

Restoration of High-Dimensional Entanglement

Once the TM is acquired, the protocol switches to a high-$K$ entangled state by removing the focusing lens, resulting in low spatial coherence and homogeneous intensity distributions. The wavefront correction pattern derived from the low-$K$ state is applied to the SLM, enabling the transmission of the high-dimensional entangled state through the scattering medium with restored quantum correlations. The second-order correlation function $G^{(2)}$ is measured to characterize the output state, with the minus-coordinate projection revealing the restoration of spatial correlations.

Figure 3: Intensity and second-order correlation images for high-Schmidt-number two-photon states under different wavefront correction conditions.

Numerical Simulations: Schmidt Number and Correction Efficiency

Numerical simulations investigate the impact of Schmidt number $K$ on speckle contrast and wavefront optimization efficiency. As $K$ increases, speckle contrast and focusing enhancement factor $\eta$ decrease, indicating reduced optimization efficacy for highly entangled states. The protocol thus relies on the ability to tune $K$ to access the regime where classical wavefront shaping techniques are effective.

Figure 4: Simulated dependence of speckle contrast and focusing enhancement on Schmidt number $K$ for two-photon states.

Implications and Future Directions

The protocol provides a resource-minimal approach to quantum wavefront correction, requiring no additional components beyond those used for entangled photon generation and shaping. It inherently matches the quantum state in all relevant degrees of freedom, circumventing mode-matching issues that plague classical beacon-based methods. This is particularly advantageous in scenarios involving spectral or polarization diversity, such as ghost imaging with undetected photons or transmission through dispersive multimode fibers.

The method is limited by the low intensity of SPDC sources and the requirement for thin phase-scattering media; extension to thick or non-planar scatterers may require multi-plane light converters (MPLC) or alternative shaping strategies. The approach is well-suited for high-dimensional quantum communication and imaging in resource-constrained or mode-diverse environments.

Conclusion

This work demonstrates a protocol for wavefront correction of high-dimensional two-photon states via coherence-entanglement transfer, enabling efficient TM measurement and correction using the quantum state itself. The method eliminates the need for classical beacons, reduces experimental overhead, and ensures optimal mode matching. The results have significant implications for the practical deployment of quantum imaging and communication protocols in complex media, with potential extensions to dynamic scatterers and multi-modal environments. Future research may focus on scaling the approach to higher photon flux, integrating advanced shaping devices, and exploring applications in quantum information processing and secure communication.