# Robustness of Entanglement for Dicke-W and Greenberger-Horne-Zeilinger Mixed States

## TL;DR

This paper studies how entanglement remains stable in mixed quantum states, comparing Dicke-W and GHZ states under different mixing ratios.

## Contribution

The paper introduces new calculations on entanglement robustness for mixed Dicke-W and GHZ states using entanglement witness methods.

## Key findings

- Entanglement robustness differs for Dicke-W and GHZ mixed states depending on mixing ratios above or below 32.
- Upper and lower bounds of entanglement robustness are calculated for mixed states with Dicke-W probabilities above or equal to 32.
- Detailed analysis of entanglement robustness is provided for mixed states with equal Dicke-W probability ratios below 32.

## Abstract

Quantum entanglement is a fundamental characteristic of quantum mechanics, and understanding the robustness of entanglement across different mixed states is crucial for comprehending the entanglement properties of general quantum states. In this paper, the robustness of entanglement of Dicke–W and Greenberger–Horne–Zeilinger (GHZ) mixed states under different mixing ratios is calculated using the entanglement witness method. The robustnesses of entanglement of Dicke–W and GHZ mixed states are different when the probability ratio of Dicke to W is greater than 32 and less than 32. For the probability of Dicke and W states greater than or equal to 32, we study the robustness of entanglement of Dicke and GHZ mixed states and analyze and calculate their upper and lower bounds. For the probability of Dicke and W states less than 32, we take the equal probability ratio of Dicke and W states as an example and calculate and analyze the upper and lower bounds of their robustness of entanglement in detail.