Entanglement polygon inequality in qudit systems

TL;DR

This paper introduces a new inequality for high-dimensional multipartite entanglement in qudit systems, providing bounds and relationships among entanglements using various entropy measures, advancing the understanding of quantum resources.

Contribution

It derives the first entanglement polygon inequality for qudit systems and extends entanglement distribution inequalities to high-dimensional cases using unified entropy measures.

Findings

Establishes bounds for three-qudit marginal entanglement.

Generalizes entanglement inequalities to various entropy types.

Provides new tools for characterizing high-dimensional entanglement.

Abstract

Entanglement is one of important resources for quantum communication tasks. Most of results are focused on qubit entanglement. Our goal in this work is to characterize the multipartite high-dimensional entanglement. We firstly derive an entanglement polygon inequality for the $q$-concurrence, which manifests the relationship among all the "one-to-group" marginal entanglements in any multipartite qudit system. This implies lower and upper bounds for the marginal entanglement of any three-qudit system. We further extend to general entanglement distribution inequalities for high-dimensional entanglement in terms of the unified-$(r, s)$ entropy entanglement including Tsallis entropy, R\'{e}nyi entropy, and von Neumann entropy entanglement as special cases. These results provide new insights into characterizing bipartite high-dimensional entanglement in quantum information processing.