Planar k -uniform states: a generalization of planar maximally entangled states

Recently, Doroudiani and Karimipour [Phys. Rev. A 102 012427(2020)] proposed the notation of planar maximally entangled states which are a wider class of multipartite entangled states than absolutely maximally entangled states. There they presented their constructions in the multipartite systems but the number of particles is restricted to be even. Here, we first solve the remaining cases, i.e., constructions of planar maximally entangled states on systems with odd number of particles. In addition, we generalized the PME to the planar k -uniform states whose reductions to any adjacent k parties along a circle of N parties are maximally mixed. We presented a method to construct sets of planar k -uniform states which have minimal support.