Tripartite entanglement of qudits

TL;DR

This paper thoroughly investigates tripartite entanglement in qudits, establishing a comprehensive framework of invariants, decomposition theorems, and classifications for complex entangled states.

Contribution

It introduces a decomposition theorem linking algebraic invariants of entanglement classes with their components, enabling efficient analysis of complex qudit entanglement.

Findings

Complete list of allowed invariant values for tripartite qudit states

Decomposition theorem relating invariants of entanglement classes and components

Explicit invariants for various irreducible entanglement classes

Abstract

We provide an in-depth study of tripartite entanglement of qudits. We start with a short review of tripartite entanglement invariants, prove a theorem about the complete list of all allowed values of three (out of the total of four) such invariants, and give several bounds on the allowed values of the fourth invariant. After introducing several operations on entangled states (that allow us to build new states from old states) and deriving general properties pertaining to their invariants, we arrive at the decomposition theorem as one of our main results. The theorem relates the algebraic invariants of any entanglement class with the invariants of its corresponding components in each of its direct sum decompositions. This naturally leads to the definition of reducible and irreducible entanglement classes. We explicitly compute algebraic invariants for several families of irreducible classes and show how the decomposition theorem allows computations of invariants for compounded classes to be carried out efficiently. This theorem also allows us to compute the invariants for the infinite number of entanglement classes constructed from irreducible components. We proceed with the complete list of the entanglement classes for three tribits with decompositions of each class into irreducible components, and provide a visual guide to interrelations of these decompositions. We conclude with numerous examples of building classes for higher-spin qudits.