Geometric approach to entanglement quantification with polynomial measures

TL;DR

This paper introduces a geometric method to quantify entanglement in rank-2 quantum states using polynomial measures, simplifying calculations and providing exact results for certain classes of states.

Contribution

It presents a novel geometric framework that relates polynomial entanglement measures to Bloch sphere geometry, enabling exact quantification for specific multi-qubit states.

Findings

Exact quantification of polynomial measures for rank-2 states.

Relation of different polynomial measures through geometric structure.

Explicit examples of three-tangle quantification for three-qubit states.

Abstract

We show that the quantification of entanglement of any rank-2 state with any polynomial entanglement measure can be recast as a geometric problem on the corresponding Bloch sphere. This approach provides novel insight into the properties of entanglement and allows us to relate different polynomial measures to each other, simplifying their quantification. In particular, unveiling and exploiting the geometric structure of the concurrence for two qubits, we show that the convex roof of any polynomial measure of entanglement can be quantified exactly for all rank-2 states of an arbitrary number of qubits which have only one or two unentangled states in their range. We give explicit examples by quantifying the three-tangle exactly for several representative classes of three-qubit states. We further show how our methods can be used to obtain analytical results for entanglement of more complex states if one can exploit symmetries in their geometric representation.