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Holographic Inequalities and Entanglement of Purification

Ning Bao, Illan F. Halpern

TL;DR

The paper advances the program of connecting holographic geometry to quantum information by rigorously examining and generalizing the conjectured $E_W=E_p$ duality between the entanglement wedge cross-section and the entanglement of purification. It introduces generalized quantities $E^G_W$ and $E^G_p$, proving upper bounds on conditional mutual information, tripartite information, and cyclic information for both, and a lower bound on tripartite information in the quantum case. It also derives a new holographic inequality for $E_W$ and provides numerical support suggesting the analogous $E_p$ inequality may hold broadly, while proposing a suboptimal-purification dictionary via bit threads. The work strengthens the case that $E_W$ and $E_p$ encode the same information-theoretic content in holographic states and lays out a concrete framework for exploring purifications beyond the minimal one and extending to covariant settings.

Abstract

We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.

Holographic Inequalities and Entanglement of Purification

TL;DR

The paper advances the program of connecting holographic geometry to quantum information by rigorously examining and generalizing the conjectured duality between the entanglement wedge cross-section and the entanglement of purification. It introduces generalized quantities and , proving upper bounds on conditional mutual information, tripartite information, and cyclic information for both, and a lower bound on tripartite information in the quantum case. It also derives a new holographic inequality for and provides numerical support suggesting the analogous inequality may hold broadly, while proposing a suboptimal-purification dictionary via bit threads. The work strengthens the case that and encode the same information-theoretic content in holographic states and lays out a concrete framework for exploring purifications beyond the minimal one and extending to covariant settings.

Abstract

We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numerical evidence that the corresponding inequality for the entanglement of purification may be true in general. Finally, we use intuition from bit threads to extend the conjecture to holographic duals of suboptimal purifications.

Paper Structure

This paper contains 20 sections, 43 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of known results
  3. Basic properties of $E_W$ and $E_p$
  4. Entanglement Entropy Inequalities
  5. Bounding holographic entanglement entropy with $E_W$
  6. Generalized $E_W$
  7. $E^G_W$ obeys known $E_W$ inequalities
  8. Upper bounding holographic conditional mutual information
  9. Upper bounding holographic tripartite information
  10. Upper bounding holographic cyclic information
  11. Cyclic $E_W$ Inequalities
  12. Bounding entanglement entropy with $E_p$
  13. Generalized $E_p$
  14. $E^G_p$ obeys known $E_p$ inequalities
  15. Upper bounding conditional mutual information
  16. ...and 5 more sections

Figures (4)

  • Figure 1: To the left, $\Gamma$ is the minimal surface the separates the entanglement wedge cross-section of $AB.$ Its area is $E_W[A:B].$ To the right, $A'$ and $B'$ purify $AB.$ For a choice of $A'$ and $B'$ over all such purifying systems that minimizes the entanglement across the dashed partition we have $E_p(A:B)=S(AA').$
  • Figure 2: Graphical proof of the upper bound on the conditional mutual information for both the case in which the regions A, B, and C are contiguous and the case in which they are disconnected. It is clear from the diagrams,and Ryu-Takayanagi, that the area of the dotted surfaces plus the area of the dash-dotted surface is greater than or equal $S(AC)$ and that the area of the dashed surfaces plus the area of the dash-dotted surface is greater than or equal $S(BC).$ Adding these two inequalities gives us the desired bound.
  • Figure 3: As depicted, for the optimal choices of $A', B'$ and $C^{(A)},$ we have $E^G_p(A:B)=S((A\backslash B)A' C^{(A)}).$
  • Figure 4: Displaying the left-hand side of Eq. (\ref{['eq:Wcheck']}) for $k=2$ and $k=3$ for $W_n$ as a function of $n,$ as well as the best fit curves of the form $D=\frac{B}{n}.$ For $k=2,$ we found $B \approx 1.537,$ and for $k=3,$ we found $B \approx 3.385.$