Entanglement polygon inequalities for pure states in qudit systems

TL;DR

This paper extends entanglement polygon inequalities to n-qudit pure states using geometric entanglement measures, explores residual entanglement in three-qubit systems, and discusses limitations and exceptions in higher-dimensional systems.

Contribution

It demonstrates the validity of entanglement polygon inequalities for n-qudit states with geometric measures and analyzes residual entanglement, revealing limitations with negativity in higher dimensions.

Findings

EPI valid for n-qudit states with geometric entanglement measure

Residual entanglement studied in three-qubit systems

Counterexamples show EPI invalid for higher dimensions with negativity

Abstract

Entanglement is one of the important resources in quantum tasks. Recently, Yang $et$ $al.$ [arXiv:2205.08801] proposed an entanglement polygon inequalities (EPI) in terms of some entanglement measures for $n$-qudit pure states. Here we continue to consider the entanglement polygon inequalities. Specifially, we show that the EPI is valid for $n$-qudit pure states in terms of geometric entanglement measure (GEM), then we study the residual entanglement in terms of GEM for pure states in three-qubit systems. At last, we present counterexamples showing that the EPI is invalid for higher dimensional systems in terms of negativity, we also present a class of states beyond qubits satisfy the EPI in terms of negativity.