For a bipartite entanglement measure <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> that satisfies the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>th-power monogamy inequality (Eq. (<InternalRef RefID="Equ1">1.1</InternalRef>)), and for its assisted counterpart <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {E}_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> that obeys the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation>th-power polygamy inequality (Eq. (<InternalRef RefID="Equ2">1.2</InternalRef>)), we introduce a unified, tunable framework indexed by a parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$m\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Within this framework, we derive two hierarchical families of refined inequalities: <OrderedList> <ListItem> <ItemNumber>(i)</ItemNumber> <ItemContent> a tightened <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-power monogamy relation for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation>, valid for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha \ge m\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mi>m</mi> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation>; </ItemContent> </ListItem> <ListItem> <ItemNumber>(ii)</ItemNumber> <ItemContent> a tightened <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-power polygamy relation for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {E}_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">E</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation>, applicable for <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(m-1)\delta < \beta \le m\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>δ</mi> <mo><</mo> <mi>β</mi> <mo>≤</mo> <mi>m</mi> <mi>δ</mi> </mrow> </math></EquationSource> </InlineEquation>. </ItemContent> </ListItem> </OrderedList> As m increases, the bounds become progressively tighter, recovering known results at <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$m=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Notably, the optimal monogamy bound emerges as a piecewise function of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, with additional correction terms activated as <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> 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.

Unified monogamy and polygamy relations for multipartite systems

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2026年2月11日

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Yue Cao,
Naihuan Jing,
Kailash Misra,
Yiling Wang

摘要

For a bipartite entanglement measure $\mathcal {E}$ $E$ that satisfies the $\gamma $ $γ$ th-power monogamy inequality (Eq. (1.1)), and for its assisted counterpart $\mathcal {E}_a$ $E_{a}$ that obeys the $\delta $ $δ$ th-power polygamy inequality (Eq. (1.2)), we introduce a unified, tunable framework indexed by a parameter $m\ge 1$ $m \geq 1$ . Within this framework, we derive two hierarchical families of refined inequalities: (i)

a tightened $\alpha $ $α$ -power monogamy relation for $\mathcal {E}$ $E$ , valid for all $\alpha \ge m\gamma $ $α \geq m γ$ ;

(ii)

a tightened $\beta $ $β$ -power polygamy relation for $\mathcal {E}_a$ $E_{a}$ , applicable for $(m-1)\delta < \beta \le m\delta $ $(m - 1) δ < β \leq m δ$ .

As m increases, the bounds become progressively tighter, recovering known results at $m=1$

m = 1

. Notably, the optimal monogamy bound emerges as a piecewise function of $\alpha $

α

, with additional correction terms activated as $\alpha $

α

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.