Completely mixed state is a critical point for three-qubit entanglement

TL;DR

This paper demonstrates that a completely mixed single-qubit state in a pure three-qubit system acts as a critical point for the geometric measure of entanglement, highlighting a unique property of tripartite quantum states.

Contribution

It establishes that when one qubit is completely mixed, the geometric entanglement measure becomes independent of polynomial invariants, identifying a critical point in three-qubit entanglement.

Findings

Geometric entanglement measure is constant when a qubit is completely mixed.

The measure is independent of polynomial invariants under these conditions.

A completely mixed state of one qubit is a critical point for entanglement measure.

Abstract

Pure three-qubit states have five algebraically independent and one algebraically dependent polynomial invariants under local unitary transformations and an arbitrary entanglement measure is a function of these six invariants. It is shown that if the reduced density operator of a some qubit is a multiple of the unit operator, than the geometric entanglement measure of the pure three-qubit state is absolutely independent of the polynomial invariants and is a constant for such tripartite states. Hence a one-particle completely mixed state is a critical point for the geometric measure of entanglement.