Automated modal analysis of entanglement with bipartite self-configuring optics

Entanglement is a unique feature of quantum mechanics. In coupled systems of light and matter, entanglement manifests itself in the linear superposition of multipartite quantum states (e.g., parametrized by the multiple spatial, spectral, or temporal degrees of freedom of a light field). In bipartite systems, the Schmidt decomposition provides a modal decomposition of the entanglement structure over independent, separable states. Although ubiquitous as a mathematical tool to describe and measure entanglement, there exists no general efficient experimental method to decompose a bipartite quantum state onto its Schmidt modes. Here, we propose a method that relies on bipartite self-configuring optics that automatically ``learns'' the Schmidt decomposition of an arbitrary pure quantum state. Our method is agnostic to the degrees of freedom over which quantum entanglement is distributed and can reconstruct the Schmidt modes and values by variational optimization of the network's output powers or coincidences. We illustrate our method with numerical examples of spectral entanglement analysis for biphotons generated via spontaneous parametric down conversion and provide experimental guidelines for its realization, including the influence of losses and impurities. Our method provides a versatile and scalable way of analyzing entanglement in bipartite integrated quantum photonic systems.