Maximally Entangled States of Four Nonbinary Particles

TL;DR

This paper introduces maximally entangled states called P-states for four nonbinary particles with three or more states, showing they are more entangled than GHZ and cluster states and classifying all such states with prime number states.

Contribution

It identifies and characterizes maximally entangled four-particle states for nonbinary systems and classifies all entanglement classes for prime-dimensional particles.

Findings

P-states exhibit stronger entanglement than GHZ and cluster states.

P-states are maximally entangled in a well-defined sense.

All entanglement classes of four-particle graph states with prime dimensions are classified.

Abstract

Systems of four nonbinary particles, each having three or more 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 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 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 a prime number (>2) of states per particle.