Characterizing quantum entanglement in mixed states is a longstanding challenge. Among the various methods available, conditional entropies serve as a powerful tool. Notably, the AR q-conditional entropy introduced by Abe and Rajagopal in 2002 has demonstrated significant promise as it often surpasses other entropy-based criteria. The wide-ranging applications of conditional entropy in quantum information underscore the importance of studying and analyzing it for a deeper understanding of quantum correlations and their implications. In this paper, we investigate the non-separability of noisy Dicke states using the AR approach of conditional entropy. Our findings reveal that the entropic criterion is equally effective as the PPT criterion in identifying non-separability across a large subset of N-partite noisy Dicke states with even N and excitation number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k = N/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mi>N</mi> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, for systems with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N > 30$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>></mo> <mn>30</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the separability thresholds derived from both criteria converge within <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$10^{-8}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>8</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, highlighting their strong agreement in this parameter range. Furthermore, we established a condition based on AR q-conditional entropy for identifying genuine multipartite entanglement (GME) in noisy Dicke states and compared its effectiveness to previous methods. Notably, our condition identifies a broader range of GME, particularly when the number of excitations approaches half the number of qubits (i.e., N/2). In contrast, previous methods perform better when the number of excitations is significantly less than N/2. We believe these results will pave the way for further advancements in entanglement theory and the development of potential quantum-based applications for conditional entropy.

Entropic analysis of non-separability in noisy Dicke states

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2025年12月2日

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Mohamed Nawareg

摘要

Characterizing quantum entanglement in mixed states is a longstanding challenge. Among the various methods available, conditional entropies serve as a powerful tool. Notably, the AR q-conditional entropy introduced by Abe and Rajagopal in 2002 has demonstrated significant promise as it often surpasses other entropy-based criteria. The wide-ranging applications of conditional entropy in quantum information underscore the importance of studying and analyzing it for a deeper understanding of quantum correlations and their implications. In this paper, we investigate the non-separability of noisy Dicke states using the AR approach of conditional entropy. Our findings reveal that the entropic criterion is equally effective as the PPT criterion in identifying non-separability across a large subset of N-partite noisy Dicke states with even N and excitation number $k = N/2$ $k = N / 2$ . Additionally, for systems with $N > 30$ $N > 30$ and $k=1$ $k = 1$ , the separability thresholds derived from both criteria converge within $10^{-8}$ $10^{- 8}$ , highlighting their strong agreement in this parameter range. Furthermore, we established a condition based on AR q-conditional entropy for identifying genuine multipartite entanglement (GME) in noisy Dicke states and compared its effectiveness to previous methods. Notably, our condition identifies a broader range of GME, particularly when the number of excitations approaches half the number of qubits (i.e., N/2). In contrast, previous methods perform better when the number of excitations is significantly less than N/2. We believe these results will pave the way for further advancements in entanglement theory and the development of potential quantum-based applications for conditional entropy.