2026/02/11 by Yue Cao, Naihuan Jing, Kailash Misra +1
Medicine · Social Sciences · Mathematics · #Mathematical and Theoretical Epidemiology and Ecology Models #Marriage and Sexual Relationships #Mathematical Inequalities and Applications
paper · pdf · doi:10.1007/s11128-026-05076-6
Abstract For a bipartite entanglement measure \mathcal E <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> that satisfies the γ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> th-power monogamy inequality (Eq. (1.1)), and for its assisted counterpart \mathcal Ea <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>a</mml:mi> </mml:msub> </mml:math> that obeys the δ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> th-power polygamy inequality (Eq. (1.2)), we introduce a unified, tunable framework indexed by a parameter m≥ 1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Within this framework, we derive two hierarchical families of refined inequalities: a tightened α <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -power monogamy relation for \mathcal E <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> , valid for all α ≥ mγ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>m</mml:mi> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> ; a tightened β <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -power polygamy relation for \mathcal Ea <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>a</mml:mi> </mml:msub> </mml:math> , applicable for (m-1)δ lt; β ≤ mδ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> <mml:mi>δ</mml:mi> <mml:mo><</mml:mo> <mml:mi>β</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>m</mml:mi> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> . As m increases, the bounds become progressively tighter, recovering known results at m=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Notably, the optimal monogamy bound emerges as a piecewise function of α <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , with additional correction terms activated as α <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> crosses successive integer thresholds, thereby offering a sharper characterization of entanglement distribution. We demonstrate that our results generalize and strengthen existing monogamy and polygamy relations through analytical comparisons and numerical evaluations using concurrence and concurrence of assistance. This hierarchical, parameterized approach offers enhanced and flexible tools for applications in quantum communication, quantum networks, and multipartite quantum information processing.