Construction of genuinely multipartite entangled subspaces and the associated bounds on entanglement measures for mixed states

Simple sufficient condition for subspace to be completely or genuinely entangled

“…Another example of a GES, which is well-known in the literature, is the antisymmetric subspace of H N,d ; this subspace is genuinely entangled because there do not exist product vectors that are anti-symmetric. Importantly, however, one should bear in mind that these subspaces only exist if d ≥ N and they are of small dimensionality, while it is known how to construct large GESs efficiently for any d and N [4] (see also [7,8]).…”

“…Quantum entanglement is one of the central notions of modern physics and technology and it has been a subject of intensive efforts in the recent decades towards its complete characterization. An important line of research in the field is the one aiming at describing properties of completely [1][2][3] or genuinely entangled subspaces [4][5][6][7][8], which are those subspaces of the composite Hilbert spaces that contain only entangled or genuinely entangled states, respectively. This was primarily motivated by their theoretical importance as any state with the support in an entangled subspaces is necessarily entangled but they are also relevant from the practical point of view, e.g., in quantum error correction [9] or in protocols where the existence of entanglement needs to be certified, for example super-dense coding [10].…”