# Self-testing of symmetric three-qubit states

## TL;DR

This paper develops device-independent self-testing schemes for a broad family of symmetric three-qubit states, including superpositions of W and GHZ states, using analytical and numerical methods.

## Contribution

It introduces new self-testing criteria for symmetric three-qubit states, extending beyond previously studied classes like Dicke and graph states.

## Key findings

- Analytical proof of self-testing for symmetric states with equal coefficients.
- Numerical self-testing of general states using SDP and swap method.
- Demonstration of high-precision self-testing for complex three-qubit states.

## Abstract

Self-testing refers to a device-independent way to uniquely identify the state and the measurement for uncharacterized quantum devices. The only information required comprises the number of measurements, the number of outputs of each measurement, and the statistics of each measurement. Earlier results on self-testing of multipartite state were restricted either to Dicke states or graph states. In this paper, we propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of W state and GHZ state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical basis, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semi-definite programming (SDP) in high precision.