Planar Maximally Entangled States

We construct a large family of planar maximally entangled (PME) states, which are a wider class of multipartite entangled states than absolutely maximally entangled (AME) states. These are states in which any half of the qudits are in a maximally mixed state, provided that they form a connected subset. We show that in contrast to AMEs, PMEs are easier to find and there are various PMEs for any even number of qudits. In particular, while it is known that no AME state of four qubits exists, we show that there are two distinct multiparameter classes of four-qubit PMEs. We also give explicit families of PMEs for any even number of particles and for any dimension. We also briefly mention the relevance of connectivity and underlying geometry to definition of a PME state. Here we have considered the simplest case of a one-dimensional graph.