Entropic analysis of non-separability in noisy Dicke states

Abstract 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 $$k = N/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Additionally, for systems with $$N &gt; 30$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>30</mml:mn> </mml:mrow> </mml:math> and $$k=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the separability thresholds derived from both criteria converge within $$10^{-8}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> </mml:msup> </mml:math> , 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.