Fully non-positive-partial-transpose genuinely entangled subspaces

TL;DR

This paper constructs genuinely entangled subspaces in multipartite systems that are fully non-positive-partial-transpose, using stabilizer formalism, and proves that such subspaces cannot support PPT entangled states.

Contribution

It introduces criteria to identify genuinely entangled stabilizer subspaces and demonstrates their maximal dimension and full NPT property.

Findings

Constructed genuinely entangled stabilizer subspaces for any number of parties.

Proved that all such subspaces are fully NPT across all bipartitions.

Showed that these subspaces cannot support PPT entangled states.

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.