Planar k-Uniform States: a Generalization of Planar Maximally Entangled States

TL;DR

This paper extends the concept of planar maximally entangled states to systems with odd numbers of particles and introduces planar k-uniform states, providing methods for their construction with minimal support.

Contribution

It generalizes PME states to odd particle systems and introduces planar k-uniform states, along with a construction method for minimal support sets.

Findings

Constructed PME states for odd particle systems.

Generalized PME to planar k-uniform states.

Presented a method for minimal support state construction.

Abstract

Recently, Doroudiani and Karimipour [Phys. Rev. A \textbf{102} 012427(2020)] proposed the notation of planar maximally entangled (PME) states which are a wider class of multipartite entangled states than absolutely maximally entangled (AME) 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.