Conditional and Multipartite Entanglements of Purification and Holography

TL;DR

The paper generalizes the entanglement of purification and its holographic dual to conditional and multipartite forms, establishing new constraints that reinforce the plausibility of the $E_p=E_W$ conjecture in conditional settings. It defines conditional $E_p$ and $E_W$, proving bounds that relate to conditional mutual information and exposing a super-Bayesian property for holographic states. It then develops multipartite versions of these quantities, proves analogous inequalities, and demonstrates that multipartite $E_p$ and $E_W$ are not holographic duals of each other, while identifying bulk-only inequalities as consistency checks. The discussion highlights implications for bulk surface partitioning, potential 2D CFT calculations, and tensor-network constructions, underscoring both the power and limits of these generalizations in holography.

Abstract

In this work we generalize the entanglement of purification and its conjectured holographic dual to conditional and multipartite versions of the same, where the optimization defining the entanglement of purification is now optimized in either a constrained way or over multiple parties. We separately derive new constraints on both the conditional entanglement of purification and its conjectured holographic dual object that match, further reinforcing the likelihood of this conjecture. We also show that the multipartite objects we define, despite obeying several of the same inequalities, are not holographic duals of each other. Further, we find inequalities that are true only for the bulk objects, and thus could provide additional consistency checks for states dual to (semi)-classical bulk geometries.

Figures (5)

• Figure 1: To the left, $A'B'$ purifies $AB.$ For a choice of $A'$ and $B'$ over all such purifying systems that minimizes the entanglement across the dashed partition we have $E_p(A:B)=S(AA').$ To the right, $\Gamma$ is the minimal surface the separates the entanglement wedge cross-section of $AB$ into a region homologous to $A$ and a region homologous to $B.$ Its area is $E_W[A:B].$ [Figure adapted with permission from figure in BaoHal].
• Figure 2: The red line is the minimal surface splitting $A_1B_1$ from $A_1B_2$ in $r(ABC)\backslash r(c),$ which is the region bounded by black lines. The RT surface of $BC$ is displayed in blue. It clearly splits the red line into two, one that splits (not necessarily minimally) $A_1$ from $A_2$ in $r(ABC)\backslash r(BC)$ and one that splits (not necessarily minimally) $B_1$ from $B_2$ in $r(BC)\backslash r(C)$.
• Figure 3: On the left, $A'B'C'$ purifies $ABC$ while minimizing $\frac{1}{3}\left(S(AA')+S(BB')+S(CC')\right).$ This minimal value is $E_p(A:B:C).$ On the right, the red surface is the minimal surface separating $r(ABC)$ into three regions, one homologous to $A,$ one to $B$ and one to $C.$ Its area is $\frac{3}{2}$ of $E_W(A:B:C).$
• Figure 4: Visual proof of the lower bound in Eq. (\ref{['eq:ulb']}) for $E_W$ shown for $k=1.$ The proof generalizes straightforwardly to arbitrary $k.$ Rearranging Eq. (\ref{['eq:ulb']}) so that terms on both sides of the inequality are all positive, we get $3 E_W(A:B:C) +S(A)+S(B)+S(C)+S(ABC) \geq S(AB)+S(AC)+S(BC).$ The black and the red lines correspond to the greater than (or equal) side of the inequality, with the red lines corresponding to the $3E_W$ term and being doubled (see definition of $E_W$ in Eq. (\ref{['eq:EWdef']})). The blue lines correspond to the lesser than (or equal) side of the inequality. The dashed black-and-blue lines appear on both sides. By subadditivity, $S(A)+S(B)+S(C) \leq S(ABC),$ allowing us to replace the red lines with the dashed black-and-blue lines. Using these and each red segment twice, one can subtend each blue arc sub-optimally.
• Figure 5: Counterexample showing that tripartite $E_p$ and $E_W$ are not dual to each other. Minimality of the RT surfaces imply that $E_W \geq \frac{1}{3} \left( S(A)+S(B)+S(C)\right),$ with the inequality being generically strict. Note, however, that a regulator is needed to makes sense of this statement since otherwise both sides of the inequality are divergent.