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Zoo of Correlation Inequalities in Holography and Beyond

Kyan Louisia, Takato Mori, Herbie Warner

TL;DR

This work systematically investigates information-theoretic inequalities for holographic correlation measures $J_W$ and $D_W$, establishing a robust topological framework that proves monotonicity, monogamy, and one-way strong superadditivity. It shows striking holographic features: $J_W$ is monogamous for both parties and obeys one-way SSA, while $D_W$ is polygamous for the unmeasured party in pure tripartite states, reflecting distinctive multipartite structures in holographic states. The authors introduce boundary proxies $J_R$ and $D_R$ via reflected entropy to obtain computable, optimization-free measures, analyzing their monotonicity and monogamy with several counterexamples. A Haar-random analysis and partial results toward two-way SSA further illuminate the landscape, and the work connects bulk geometry to classical/quantum correlations and distillable entanglement, framing a comprehensive “zoo” of inequalities in holography and beyond.

Abstract

We study information-theoretic inequalities for holographic correlation measures $J_W$ and $D_W$, establishing rigorous topological frameworks that prove monotonicity, monogamy, and strong superadditivity. While earlier work proposed bulk duals of classical correlation and quantum discord, their information-theoretic properties were unclear. We derive multiple inequalities involving entanglement wedge cross sections (EWCS) and show that monotonicity holds for both measured and unmeasured parties in $J_W$ and for the unmeasured party in $D_W$. Unlike their original counterparts, we find that $J_W$ is monogamous for both parties and obeys one-way strong superadditivity, whereas $D_W$ is polygamous for the unmeasured party in pure tripartite states. These results highlight distinctive features of holographic states and support a conjectured duality between $J_W$ and distillable entanglement. Motivated by the relation between EWCS and reflected entropy, we introduce boundary analogues $J_R$ and $D_R$, which serve as computable proxies for classical and quantum correlations without optimization. We analyze their inequalities, proving several and presenting counterexamples. Overall, we provide a systematic "zoo" of correlation inequalities in and beyond holography, clarifying connections among bulk geometry, discord-type correlations, and distillable entanglement.

Zoo of Correlation Inequalities in Holography and Beyond

TL;DR

This work systematically investigates information-theoretic inequalities for holographic correlation measures and , establishing a robust topological framework that proves monotonicity, monogamy, and one-way strong superadditivity. It shows striking holographic features: is monogamous for both parties and obeys one-way SSA, while is polygamous for the unmeasured party in pure tripartite states, reflecting distinctive multipartite structures in holographic states. The authors introduce boundary proxies and via reflected entropy to obtain computable, optimization-free measures, analyzing their monotonicity and monogamy with several counterexamples. A Haar-random analysis and partial results toward two-way SSA further illuminate the landscape, and the work connects bulk geometry to classical/quantum correlations and distillable entanglement, framing a comprehensive “zoo” of inequalities in holography and beyond.

Abstract

We study information-theoretic inequalities for holographic correlation measures and , establishing rigorous topological frameworks that prove monotonicity, monogamy, and strong superadditivity. While earlier work proposed bulk duals of classical correlation and quantum discord, their information-theoretic properties were unclear. We derive multiple inequalities involving entanglement wedge cross sections (EWCS) and show that monotonicity holds for both measured and unmeasured parties in and for the unmeasured party in . Unlike their original counterparts, we find that is monogamous for both parties and obeys one-way strong superadditivity, whereas is polygamous for the unmeasured party in pure tripartite states. These results highlight distinctive features of holographic states and support a conjectured duality between and distillable entanglement. Motivated by the relation between EWCS and reflected entropy, we introduce boundary analogues and , which serve as computable proxies for classical and quantum correlations without optimization. We analyze their inequalities, proving several and presenting counterexamples. Overall, we provide a systematic "zoo" of correlation inequalities in and beyond holography, clarifying connections among bulk geometry, discord-type correlations, and distillable entanglement.

Paper Structure

This paper contains 64 sections, 14 theorems, 216 equations, 19 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classical and Quantum Correlations
  3. Classical and Quantum Correlation Measures
  4. Mutual Information.
  5. Classical Correlation.
  6. Quantum Discord.
  7. Holographic duals
  8. Holographic entanglement entropy
  9. Entanglement wedge cross section
  10. Bulk duals for classical and quantum correlations
  11. Reflected entropy
  12. One-shot distillable entanglement.
  13. Correlation Inequalities: Monotonicity and Monogamy
  14. Example
  15. Assumptions and Technical Tools
  16. ...and 49 more sections

Key Result

Lemma 3.1

This is a standard property of holographic spacetimes under our other assumptions where any two distinct length minimizing geodesics on the slice intersect in the interior of the slice at most once. If they intersect more than once, say at points $x$ and $y$, then they coincide between $x$ and $y$.

Figures (19)

  • Figure 1: Example of definitions given below on the Poincaré disk.
  • Figure 2: Decomposition of $\mathcal{E}(AB)$ into its $A$-only component in red, $B$-only component in orange and a disjoint bridged entanglement wedge $\mathcal{E}_{\text{brid}}(A:B)$ in blue. We have $\{A_5,A_6\}$ are $A$-only components of $AB$, $B_4$ a $B$-only component of $AB$, and $\{A_1,A_2,A_3,A_4,B_1,B_2,B_3\}$ bridged components of $AB$.
  • Figure 3: Example case of \ref{['eq: simple inequality to prove']}. On the LHS blue curves are $\gamma_{AB}$ and red curves $\Gamma^W_{A:B}$. On the RHS they are $\gamma_A$ and $\gamma_B$ respectively. We define $A = \cup A_i$ and $B = \cup B_i$. Topologies of entanglement wedges chosen to exemplify cases of proof below.
  • Figure 4: Decomposition of Figure \ref{['Fig: structure of proofs']} into $A$ homologous regions. We compute the closure via equation \ref{['eq: closure of ua']} which just glues $\Gamma^W_{A:B}$ to $U_A$. Area bound via $\geq$ computed over the area of the boundary of each region.
  • Figure 5: Dashed lines are $\gamma_{AB}$, the red line $\Gamma$ is some surface we are checking for $(A:B)$ EWCS admissibility, and the blue some simple path $\ell$. On the LHS $\Gamma$ is EWCS admissible as all choices of $\ell$ intersect $\Gamma$ when traveling from $A$ to $B$ within $\mathcal{E}(AB)$. The RHS is not however as $\ell$ can start adjacent to $A$ and reach $B$ without crossing $\Gamma$.
  • ...and 14 more figures

Theorems & Definitions (43)

  • Definition 2.1: Monotonicity
  • Definition 2.2: Monogamy
  • Definition 2.3: Strong superadditivity
  • Lemma 3.1: No multiple intersections of geodesics on an AdS Cauchy slice
  • Definition 3.1: Bulk
  • Definition 3.2: Boundary notation
  • Definition 3.3: Entanglement wedge
  • Definition 3.4: Bridged wedges
  • Remark 3.1: Bridged wedges and correlation measures
  • Definition 3.5: Homology
  • ...and 33 more