# Symmetric 3 Qubit State Invariants

## TL;DR

This paper characterizes the set of possible values for three key entanglement measures in pure symmetric 3-qubit states, providing a clear understanding of their interrelations.

## Contribution

It derives the explicit achievable region of the three entanglement invariants for symmetric 3-qubit states using a canonical form from Majorana representation.

## Key findings

- Explicit region of entanglement invariants identified
- Relationship between pairwise concurrence, 3-tangle, and Kempe invariant clarified
- Provides a geometric understanding of symmetric 3-qubit entanglement measures

## Abstract

For pure symmetric 3-qubit states there are only three algebraically independent entanglement measures; one choice is the pairwise concurrence $\mathcal C$, the 3-tangle $\tau$, and the Kempe invariant $\kappa$. Using a canonical form for symmetric $N$-qubit states derived from their Majorana representation, we derive the explicit achievable region of triples $(\mathcal C,\tau,\kappa)$.