Multipartite Reflected Entropy

TL;DR

This work extends the holographic notion of reflected entropy to arbitrary $n$-party states by constructing two bulk geometries whose minimal surfaces compute multiples of the $n$-party entanglement wedge cross-section $E_W$. It employs replica gluing and canonical purification to produce bulk spacetimes whose reflected minimal surface areas equal $S_R=2E_W$ (and $S_R=4E_W$ in certain parity cases), and provides two distinct topologies (Candidates 1 and 2) with different boundary interpretations. In AdS$_3$/CFT$_2$, the authors interpret the bulk constructions as multiboundary wormholes, linking multipartite entanglement to boundary topology while noting generalization challenges to higher dimensions. They also discuss practical aspects such as $E_W$ winding, higher-dimensional extensions, and the potential for using $S_R$ as a tractable proxy to study $E_P$ dynamics in holographic settings.

Abstract

We discuss two methods that, through a combination of cyclically gluing copies of a given $n$-party boundary state in AdS/CFT and a canonical purification, creates a bulk geometry that contains a boundary homologous minimal surface with area equal to 2 or 4 times the $n$-party entanglement wedge cross-section, depending on the parity of the party number and choice of method. The areas of the minimal surfaces are each dual to entanglement entropies that we define to be candidates for the $n$-party reflected entropy. In the context of AdS$_3$/CFT$_2$, we provide a boundary interpretation of our construction as a multiboundary wormhole, and conjecture that this interpretation generalizes to higher dimensions.

Figures (7)

• Figure 1: An example of the $E_W$ surface for a 3-party boundary state. The homologous surfaces that form the $E_W(A:B:C)$ surface are shown in solid lines. The 3 corresponding surfaces $\tilde{A},\tilde{B},\tilde{C}$ are shown in dashed lines, containing their respective boundary regions, as well as part of the minimal surfaces that separate the boundary regions.
• Figure 2: The bulk interpretation of the canonical purification for a 3-party boundary state. The geometry is doubled and glued to the original geometry along identical minimal surfaces that separate the boundary regions. The result is a "pair of pants" topology with 3 asymptotic regions. From a boundary perspective, this geometry corresponds to a 3-boundary wormhole in the context of AdS$_3$/CFT$_2$. As we note below, simply doubling the original geometry leads to a reflected entropy defined on the glued geometry will generically fail to capture multipartite correlations.
• Figure 3: An example of the gluing construction for $n=4$ with the assumptions made in the text. Left: The arrows indicate the surfaces that are identified in the gluing. The red lines inside the copies indicate the multipartite $E_W$ surfaces for each boundary region. The sum of the areas in Planck units of these surfaces gives $E_W(A:B:C:D)$. Right: After the gluing and the canonical purification, we have a ring that is punctured in various locations. The reflected minimal surface consists of two disconnected pieces, one on the outside of the ring and one on the inside, that each have area $E_W(A:B:C:D)$.
• Figure 4: An example of the second candidate for the multipartite reflected entropy for $n=4$. The left figure shows the construction of the glued geometry, and the right figure is the final purified geometry. The $E_W$ surfaces are shown in red. The final glued, purified geometry no longer has the ring-like shape from the first candidate. The reflected minimal surface is one connected piece with area $2E_W(A:B:C:D)$. On comparison to figure \ref{['fig:n4-gluing']}, it is also clear that this construction has broken the cyclical symmetry of the boundary regions.
• Figure 5: Comparing the two constructions for $n=3$. Above: For Candidate 1, we need 6 copies for the replica step, which is then mirrored for a total of 12. The topology is ring-like, and the reflected minimal surface consists of 2 disjoint surfaces, each of which has area equal to $2E_W(A:B:C)$. The reflected entropy is $S_R(A:B:C) = S[(A^6A^1A^2A^{6\star}A^{1\star}A^{2\star})(B^4B^5B^6B^{4\star}B^{5\star}B^{6\star})(C^2C^3C^4C^{2\star}C^{3\star}C^{4\star})]$. Below: For Candidate 2, we only need 3 copies to do the replica step, which is then mirrored for a total of 6. The red lines denote the $E_W$ surfaces on each surface, which combine to form the reflected minimal surface with area $2E_W(A:B:C)$. The reflected entropy is $S_R(A:B:C) = S[(A^1A^2A^{1\star}A^{2\star})(C^2C^3C^{2\star}C^{3\star})]$.