The nine ways of four qubit entanglement and their threetangle

TL;DR

This paper calculates the mixed threetangle for four-qubit states, compares methods for estimating entanglement measures, and provides new bounds, enhancing understanding of multipartite entanglement structures.

Contribution

It introduces a new upper bound for the mixed threetangle in four-qubit states and compares it with existing methods, improving entanglement quantification techniques.

Findings

New upper bound for mixed threetangle in most cases

Comparison shows the convex roof often matches the upper bound

Identifies conditions where the convex roof is exactly obtained

Abstract

I calculate the mixed threetangle $\tau_3[\rho]$ for the reduced density matrices of the four-qubit representant states found in Phys. Rev. A {\bf 65}, 052112 (2002). In most of the cases, the convex roof is obtained, except for one class, where I provide with a new upper bound, which is assumed to be very close to the convex roof. I compare with results published in Phys. Rev. Lett. {\bf 113}, 110501 (2014). Since the method applied there usually results in higher values for the upper bound, in certain cases it can be understood that the convex roof is obtained exactly, namely when the zero-polytope where $\tau_3$ vanishes shrinks to a single point.