$X$-States From a Finite Geometric Perspective

TL;DR

This paper classifies two-qubit X-states based on their entanglement properties, linking their characteristics to geometric hyperplanes in symplectic polar spaces, and introduces hyperplane-states with non-local features.

Contribution

It provides a geometric classification of X-states using symplectic polar spaces and introduces the novel concept of hyperplane-states.

Findings

15 types of X-states split into two entanglement-based sets

Characterization of X-states with maximally-mixed subsystems

Introduction of hyperplane-states and their non-local properties

Abstract

It is found that $15$ different types of two-qubit $X$-states split naturally into two sets (of cardinality $9$ and $6$) once their entanglement properties are taken into account. We {characterize both the validity and entangled nature of the $X$-states with maximally-mixed subsystems in terms of certain parameters} and show that their properties are related to a special class of geometric hyperplanes of the symplectic polar space of order two and rank two. Finally, we introduce the concept of hyperplane-states and briefly address their non-local properties.