Exact zeros of entanglement for arbitrary rank-two mixtures: how a geometric view of the zero-polytope makes life more easy

TL;DR

This paper introduces a geometric method using the zero-polytope and Bloch sphere analogy to exactly determine zeros of entanglement measures for rank-two mixed states, simplifying complex calculations.

Contribution

It presents a novel geometric approach to calculate zeros of entanglement measures, applicable to arbitrary polynomial degrees and local dimensions, demonstrated on three-tangle and superpositions of GHZ and W states.

Findings

Exact determination of zero-polytope for three-tangle.

Upper bounds to convex roof are below linearized bounds.

Method applies to arbitrary polynomial entanglement measures.

Abstract

Here I present a method how intersections of a certain density matrix of rank two with the zero-polytope can be calculated exactly. This is a purely geometrical procedure which thereby is applicable to obtaining the zeros of SL- and SU-invariant entanglement measures of arbitrary polynomial degree. I explain this method in detail for a recently unsolved problem. In particular, I show how a three-dimensional view, namely in terms of the Boch-sphere analogy, solves this problem immediately. To this end, I determine the zero-polytope of the three-tangle, which is an exact result up to computer accuracy, and calculate upper bounds to its convex roof which are below the linearized upper bound. The zeros of the three-tangle (in this case) induced by the zero-polytope (zero-simplex) are exact values. I apply this procedure to a superposition of the four qubit GHZand W-state. It can however be applied to every case one has under consideration, including an arbitrary polynomial convex-roof measure of entanglement and for arbitrary local dimension.