Manifolds of Product States of Three Qubits

Quantum entanglement has played a pivotal role in both theoretical investigations and practical applications within quantum information science. In this study, we explore the connection between entanglement and geometric structures, specifically manifolds and their associated geometric properties such as curvature. We focus on the manifolds formed by the product states of three qubits, examining the induced metric derived from the Euclidean metric, the Levi-Civita connection, and, where computationally tractable, the scalar curvature. Consequently, separable states can be characterized as convex combinations of points residing on these manifolds, whereas non-separable states exhibit entanglement.