Maximally entangled states of four nonbinary particles

Systems of four nonbinary particles, with each particle having $d\ge 3$ 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 $d\ge 3$.

Figure 1

Graphs defining the four-particle GHZ ($G$), cluster ($C$), and $P$ states. Blue (light gray) edges dictate ${\omega }^{ij}$ factors, and red (dark gray) edges dictate ${\omega }^{-ij}$.

Figure 2

Number of measurement paths per qutrit for each sequence of outcomes.

Figure 3

Most general six-edged graph and its reduction to an equivalent five-edged graph.