Generalizations of Reflected Entropy and the Holographic Dual

TL;DR

This work extends the concept of reflected entropy to multipartite states by introducing a family of generalized reflected entropies, with a central permutation-invariant member $Δ_R$ defined via successive canonical purifications. It establishes a holographic dictionary linking these quantities to minimal surfaces formed from entanglement-wedge cross sections, and provides strong evidence that, in AdS$_3$/CFT$_2$, $Δ_R$ equals $2Δ_W$ for tripartite configurations. The authors perform an explicit large-$c$ calculation using replica trick and twist operators, finding agreement with the holographic computation, thereby reinforcing the proposed duality. The results illuminate how multipartite entanglement structures encode bulk geometry and pave the way for systematic generalizations to higher $n$-partite cases and higher-dimensional holographic correspondences.

Abstract

We introduce a new class of quantum and classical correlation measures by generalizing the reflected entropy to multipartite states. We define the new measures for quantum systems in one spatial dimension. For quantum systems having gravity duals, we show that the holographic duals of these new measures are various types of minimal surfaces consist of different entanglement wedge cross sections. One special generalized reflected entropy is $Δ_R$, with the holographic dual proportional to the so called multipartite entanglement wedge cross section $Δ_W$ defined before. We then perform a large $c$ computation of $Δ_R$ and find precise agreement with the holographic computation of 2$Δ_{W}$. This agreement shows another candidate $Δ_R$ as the dual of $Δ_W$ and also supports our holographic conjecture of the new class of generalized reflected entropies.

Figures (14)

• Figure 2.1: Tripartite entanglement wedge cross-sections $\Delta_W$ of subsystems $ABC$ in AdS$_3$/CFT$_2$. $\mathit{Left}:$ A pure state in $2d$ CFT on a circle made up of six intervals: $A,B,C,a,b\text{ and }c$. The dotted lines denote Ryu-Takayanagi surfaces of $ABC$. $\mathit{Right}:$ Entanglement wedge, the interior of the Ryu-Takayanagi surfaces $\cup\ ABC$, in which the closed curve denotes $\Delta_W$.
• Figure 2.2: Canonical purification of $\rho_{AB}$: $|\sqrt{\rho_{AB}}\rangle = |\sqrt{\text{Tr}_c|\psi\rangle\langle\psi|}\rangle$. Tracing out $c$ corresponds to gluing $c$ from 2 circles and we view this process as a fundamental step to obtain a big pure state. The red dashed line separates $AA^*$ from $BB^*$ and defines reflected entropy $S_R$.
• Figure 2.3: The procedure to construct the pure state with three similar steps. Step $i$: from the original pure state $\rho_0=|\psi_{ABCabc}\rangle \langle \psi_{ABCabc}|$ to $\psi_1=|\sqrt{\text{Tr}_c\rho_0}\,\rangle$. Step $ii$: from $\rho_1=|\psi_1\rangle \langle \psi_1|$ to $\psi_2=|\sqrt{\text{Tr}_{bb'}\rho_1}\,\rangle$. Step $iii$: from $\rho_2=|\psi_2\rangle \langle \psi_2|$ to $\psi_3=|\sqrt{\text{Tr}_{aa'a"a"'}\rho_2}\,\rangle$, whose density matrix is $\rho_3=|\psi_3\rangle \langle \psi_3|$ and this is the boundary state in final 8-copy purification (also seen in Fig.\ref{['ps']}).
• Figure 3.1: Canonical purification of $\rho_{AB}$ together with entanglement wedges: Tracing out $c$ corresponds to gluing Ryu-Takayanagi surfaces (blue lines) for two copies of entanglement wedges and the new bulk geometry describes two entangled boundary quantum systems $AA^*$ and $BB^*$. We view this process as a fundamental step to obtain a bulk geometry describing a big pure state. The orange line is the minimal surface in the bulk seperating $AA^*$ from $BB^*$.
• Figure 3.2: A pure state constructed by 8 copies of the subsystem $ABC$ together with dual glued bulk. The entangling surface (denoted by the closed orange curve) is just twice of the minimal cross sections in Fig.\ref{['ew']}, which is also the holographic dual of entanglement entropy $S(AA'A_1A'_1B_1B'_1B_{1'}B'_{1'}CC_{'}C_1C_{1'}:A_{'}A'_{'}A_{1'}A'_{1'}BB'B_{'}B'_{'}C'C'_{'}C'_1C'_{1'})$, defined to be multipartite reflected entropy of subsystems $ABC$, namely $\Delta_R(A:B:C)$. It can be seen that $\Delta_R(A:B:C)=2\Delta_W(A:B:C)$ for holographic states.