Entanglement Polygon Inequality in Qubit Systems

TL;DR

This paper establishes tight inequalities governing the distribution of entanglement in multi-qubit systems, revealing geometric and sharing constraints that deepen understanding of quantum entanglement structure.

Contribution

It introduces a set of tight entanglement inequalities for N-qubit pure states, providing bounds and sharing properties using a geometric polytope approach.

Findings

Derived tight bounds for bipartite entanglements

Illustrated inequalities with GHZ and W states

Revealed geometric structure of entanglement sharing

Abstract

We prove a set of tight entanglement inequalities for arbitrary $N$-qubit pure states. By focusing on all bi-partite marginal entanglements between each single qubit and its remaining partners, we show that the inequalities provide an upper bound for each marginal entanglement, while the known monogamy relation establishes the lower bound. The restrictions and sharing properties associated with the inequalities are further analyzed with a geometric polytope approach, and examples of three-qubit GHZ-class and W-class entangled states are presented to illustrate the results.