Description
We introduce an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Our method utilizes a modified Veronese embedding, restricted to the conic, to generate a subspace within the symmetric subspace of the Hilbert space. Using this method, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), which enables the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). Furthermore, we extend this framework by employing a Veronese embedding for multipartite systems and systematically investigate the transition from Veronese to Segre-Veronese embeddings by imposing constraints on the affine coordinates of the points, which, in turn, increases the dimension of the CES while reducing the dimension of the GES.
