Fundamental Bounds of Wavefront Shaping of Spatially Entangled Photons

Abstract: Wavefront shaping enables control of classical light through scattering media. Extending these techniques to spatially entangled photons promises new quantum applications, but their fundamental limits, especially when both photons scatter, remain unclear. Here, we theoretically and numerically investigate the enhancement of two-photon correlations through thick scattering media. We analyze configurations where a spatial light modulator shapes one or both photons, either before or after the medium, and show that the optimal enhancement differs fundamentally from classical expectations. For a system with $N$ modes, we show that shaping one photon yields the classical enhancement $\eta \approx (\pi/4)N$, while shaping both photons before the medium reduces it to $\eta \approx (\pi/4)^2N$. However, in some symmetric detection schemes, when both photons are measured at the same mode, perfect correlations are restored with $\eta \approx N$, resembling digital optical phase conjugation. Conversely, shaping both photons after the medium leads to a complex, NP-hard-like optimization problem, yet achieves superior enhancements, up to $\eta \approx 4.6N$. These results reveal unique quantum effects in complex media and identify strategies for quantum imaging and communication through scattering environments.