Learning the Geometry of Quantum Resources: Entanglement, Magic, and Entropy Cones

Entanglement and magic serve as complementary resources that determine the computational capabilities of quantum computers. Entanglement admits a geometric interpretation through entropy vectors, which capture subsystem correlations and are constrained by families of entropy inequalities. Quantum magic, a resource essential for quantum advantage, provides an additional state classification and plays a role in identifying violations of certain entropy inequalities. In this work, we introduce a machine learning framework to analyze the coupled dynamics of entanglement and magic in quantum circuits. We develop an efficient algorithm for generating states with specific entanglement and magic characteristics, those which yield violations of desired entropy inequalities, and construct circuits to prepare such states. This framework enables a systematic exploration of the nested structure of entropy cones, and the transitions between them. Moreover, our protocol enables the informed design of quantum algorithms with tailored information-theoretic properties. Our combined mathematical and computational techniques strengthen the connection between quantum information, emergent phenomena, and the role of quantum resources in high-energy physics and holography.