Generalized X states of N qubits and their symmetries

TL;DR

This paper introduces a comprehensive family of N-qubit X states that unify various known states, providing algebraic tools and entanglement detection methods, with connections to projective geometry.

Contribution

It generalizes X states to N qubits, develops their algebraic structure, and demonstrates entanglement detection techniques, extending prior two-qubit frameworks.

Findings

Unified N-qubit X state family with 2^(N+1)-1 parameters

Algebraic characterization and iterative construction of state operators

Entanglement witnesses effectively detect entanglement in these states

Abstract

Several families of states such as Werner states, Bell-diagonal states and Dicke states are useful to understand multipartite entanglement. Here we present a [2^(N+1)-1]-parameter family of N-qubit "X states" that embrace all those families, generalizing previously defined states for two qubits. We also present the algebra of the operators that characterize the states and an iterative construction for this algebra, a sub-algebra of su(2^(N)). We show how a variety of entanglement witnesses can detect entanglement in such states. Connections are also made to structures in projective geometry.