Fully non-positive-partial-transpose genuinely entangled subspaces

Abstract

Genuinely entangled subspaces are a class of subspaces in the multipartite Hilbert spaces that are composed of only genuinely entangled states. They are thus an interesting object of study in the context of multipartite entanglement. Here we provide a construction of multipartite subspaces that are not only genuinely entangled but also fully non-positive-partial-transpose (NPT) in the sense that any mixed state supported on them has non-positive partial transpose across any bipartition. Our construction originates from the stabilizer formalism known for its use in quantum error correction. To this end, we first introduce a couple of criteria allowing to assess whether any state from a given non-trivial stabilizer subspace is genuinely multipartite entangled. We then use these criteria to construct genuinely entangled stabilizer subspaces for any number of parties and arbitrary local dimension and conjecture them to be of maximal dimension achievable within the stabilizer formalism. At the same time, we prove that every genuinely entangled subspace is fully NPT in the above sense, which implies a quite surprising fact that no genuinely entangled stabilizer subspace can support PPT entangled states.

Universal construction of genuinely entangled subspaces of any size

“…A somewhat complementary question is whether our approach allows for a construction of fully nonpositive partial transpose (NPT) GESs, that is GESs that support only states being NPT across all bipartitions [9,23], and in case of a positive answer which dimensions of GESs could be achieved. This is especially intriguing in view of the recent construction of large fully NPT stabilizer GESs [21]. We leave both problems for future study as it seems that without a fairly simple explicit orthogonal bases for the subspaces both problems may turn out analytically formidable.…”

“…[7] was based on unextendible product bases (UPBs) [15,16] a natural notion for constructing entangled sub-Maciej Demianowicz: maciej.demianowicz@pg.edu.pl spaces -specifically, their nonorthogonal variant (nUPBs) [17]. Further developments made use of various tools and concepts: unextendible biproduct bases [9], certain characterization of bipartite completely entangled subspaces [18], stabilizer formalism [19,20,21], correspondence between quantum channels and subspaces of tensor product Hilbert spaces [22], or compositional (tensor product) approach [23]. While some of these proposal [18,22] offer the possibility of constructing GESs of any sizes, their actual general utility is only theoretical as they do not provide a recipe to achieve this task in any multipartite scenario and their applicability is in fact very limited and boils down to small systems.…”