Reflected multi-entropy and its holographic dual

TL;DR

The paper introduces reflected multi-entropy as a UV-defined multipartite mixed-state measure via canonical purification and proposes its holographic dual as a minimal surface web. For the tripartite case, it computes $S_R^{(3)}$ using a six-point twist-operator correlator in a large-$c$ CFT and compares with a holographic computation based on a bulk minimal-web length, finding precise agreement at zero and finite temperature. This work extends the AdS/CFT dictionary to a new class of multipartite mixed-state correlations, providing concrete computational tools (monodromy analysis, BTZ backgrounds) and highlighting the role of OPE data in fixing the correlators. The results suggest a robust framework for diagnosing multipartite entanglement in mixed states and its bulk dual, with potential implications for chaos, thermalization, and topological order in holographic systems.

Abstract

We introduce a mixed-state generalization of the multi-entropy through the canonical purification, which we call reflected multi-entropy. We propose the holographic dual of this measure. For the tripartite case, a field-theoretical calculation is performed using a six-point function of twist operators at large $c$ limit. At both zero and finite temperature, the field-theoretical results match the holographic results, supporting our holographic conjecture of this new measure.

Figures (11)

• Figure 1: The construction of the $n^{\mathtt{q} - 1}$-sheet Riemann surface in the special case $\mathtt{q} = 3$ and $n = 2$.
• Figure 2: Canonical purification of $\rho_{ABC}$. (a) Global pure state $\psi_{AcBaCb}$ defined on a circle. Tracing out $abc$ gives a mixed state $\rho_{ABC}$. (b) Picking up another copy of $\rho_{ABC}$ and gluing these two copies along $abc$ gives a big pure state $\ket{\sqrt{\rho_{ABC}}}$\ref{['eq-canonicalP']}, which is the canonical purification of $\rho_{ABC}$.
• Figure 3: (a) Canonical purification of $\rho_{ABC}$ and its holographic dual. Tracing out $abc$ corresponds to gluing along the RT surface for $\rho_{ABC}$ (blue curves). The orange web $\mathcal{W}$ is the minimal web in the bulk as the holographic dual of $S^{(3)}(AA^*;BB^*;CC^*)$. (b) The single copy picture. The shadow area is the entanglement wedge of $A\cup B\cup C$ and the orange web is the holographic dual of $S_R^{(3)}(A;B;C)$.
• Figure 4: The holographic dual of $S^{(\mathtt{q} = 3)}_R(A;B;C)$.
• Figure 5: Comparison of the partial derivatives of $\frac{c}{3} L[\mathcal{W}]$ and $S_R^{(\mathtt{q} = 3)}$ (divided by $c/3$) with respect to $\xi = a,b,r$ at zero temperature. We take $a = 20$, $b = 100$, and $r$ ranging from $1$ to $10$. The holographic results and the field-theoretical results match well (the difference between them is within 0.4%).