Entanglement in cyclic sign invariant quantum states

We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite properties, such as positivity and positivity under partial transpose (PPT), can be simply characterized in terms of these vectors and their discrete Fourier transforms. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d ≤ 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from from the literature. Our results show that these states have a rich entanglement structure, even in the bosonic subspace.