Quantifying subspace entanglement with geometric measures

Abstract

Determining whether a subspace spanned by certain quantum states is entangled and its entanglement dimensionality remains a fundamental challenge in quantum information science. This paper introduces a geometric measure of $r$-bounded rank, $E_r(\mathcal{S})$, for a given subspace $\mathcal{S}$. Derived from the established geometric measure of entanglement, this measure is specifically designed to assess the entanglement within $\mathcal{S}$. It not only serves as a tool for determining the entanglement dimensionality but also illuminates the subspace's capacity to preserve such entanglement. By employing developed non-convex optimization techniques utilized in machine learning area, we can accurately calculate $E_r(\mathcal{S})$ within the manifold optimization framework. Our approach demonstrates notable advantages over existing hierarchical methods, PPT relaxation techniques, and the seesaw strategy, particularly by combining computational efficiency with broad applicability. More importantly, it paves the way for high-dimensional entanglement certification, which is crucial for numerous quantum information tasks. We showcase its effectiveness in validating high-dimensional entangled subspaces in bipartite systems, determining the border rank of multipartite pure states, and identifying genuinely or completely entangled subspaces.