Geometric representation of two-qubit entanglement witnesses

TL;DR

This paper extends the geometric representation of two-qubit states to entanglement witnesses using steering ellipsoids, providing a new classification scheme based on block positivity and revealing properties like witness optimality visually.

Contribution

It introduces a geometric framework for representing and classifying two-qubit entanglement witnesses via steering ellipsoids, expanding the understanding of their properties.

Findings

Ellipsoids inside the Bloch sphere represent states or entanglement witnesses.

Classification scheme based on determinants of B and its partial transpose.

Properties like witness optimality are visually manifest in the geometric representation.

Abstract

Any two-qubit state can be represented geometrically by a steering ellipsoid inside the Bloch sphere. We extend this approach to represent any block positive two-qubit operator $B$. We derive a novel classification scheme based on the positivity of $\det B$ and $\det B^\mathrm{T_B}$; this shows that any ellipsoid inside the Bloch sphere must represent either a two-qubit state or a two-qubit entanglement witness. We focus on such witnesses and their corresponding ellipsoids, finding that properties such as witness optimality are naturally manifest in this geometric representation.