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

• Published in: Entropy
• Main Authors: Ling-Hui Zhu, Zhen Zhu, Guo-Lin Lv, Chong-Qiang Ye, Xiao-Yu Chen
• Format: Article
• Language: English
• Published: MDPI AG 2024-09-01
• Subjects: quantum entanglement; robustness of entanglement; Dicke state; W state; GHZ state
• Online Access: https://www.mdpi.com/1099-4300/26/9/804

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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></semantics></math></inline-formula> and less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></semantics></math></inline-formula>. For the probability of Dicke and W states greater than or equal to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></semantics></math></inline-formula>, 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></semantics></math></inline-formula>, 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.