We describe the entanglement of two indistinguishable delocalized spin-
21 particles in the simplest spatial configuration of three spatial modes with the constraint that at most one particle occupy each mode. It is shown that this is the only number of modes for which maximally entangled states exist in such a system. The set of entangled states, including the set of maximally entangled states, is described and different types of entanglement in terms of Bell-nonlocal correlations for different partitions of the system are identified. In particular we focus on the entangled states that are Bell local for a tripartition of the system and cannot be described as a superposition of bipartite entangled pairs of localized particles. Two entanglement invariants are constructed and it is shown that all entanglement monotones are functions of these. Furthermore, the system has a generic nontrivial local unitary symmetry with a corresponding $2\ensuremath{\pi}/3$ fractional topological phase. In addition to this a necessary and sufficient condition for the existence of maximally entangled states in systems of arbitrary numbers of delocalized particles with arbitrary spin, where at most one particle can occupy each mode, is derived.
Systems of four nonbinary particles, with each particle having $d\ensuremath{\ge}3$ internal states, exhibit maximally entangled states that are inaccessible to four qubits. This breaks the pattern of two- and three-particle systems, in which the existing graph states are equally accessible to binary and nonbinary systems alike. We compare the entanglement properties of these special states (called
P states) with those of the more familiar Greenberger-Horne-Zeilinger (GHZ) and cluster states accessible to qubits. The comparison includes familiar entanglement measures, the ``steering'' of states by projective measurements, and the probability that two such measurements, chosen at random, leave the remaining particles in a Bell state. These comparisons demonstrate not only that
P-state entanglement is stronger than the other types but also that it is maximal in a well-defined sense. We prove that GHZ, cluster, and
P states represent all possible entanglement classes of four-particle graph states with prime $d\ensuremath{\ge}3$.
In entanglement theory, there are different methods to consider one state being more entangled than another. The “maximally” entangled states in a multipartite system can be defined from an axiomatic perspective. According to different criteria for selection, there are many specific types of quantum maximally entangled states, such as absolutely maximally entangled state, planar maximally entangled state and so on. In this paper we propose a new type of maximally entangled states, the planar two-region multipartite maximally entangled state. The requirement condition of this maximally entangled state is weak than that of the absolutely maximally entangled state and different from that of the planar maximally entangled state. We show that there are the two-region four-partite maximally entangled states in 4-qubit and 7-qubit planar systems, although there is no absolutely maximally entangled state in these systems. It is proved that there are the planar two-region four-partite maximally entangled states in both even particle quantum systems and odd particle quantum systems. Additionally, based on some planar two-region four-partite maximally entangled states, the new planar two-region four-partite maximally entangled states are generated. We also provide some important examples of the planar two-region multi-partite maximally entangled states.