A canonical purification for the entanglement wedge cross-section

TL;DR

The paper defines and analyzes the reflected entropy S_R as the entanglement entropy of AA^* in the canonical purification of ρ_AB, and demonstrates a robust holographic dual to the entanglement wedge cross-section E_W. It develops a multifaceted toolkit—geometric constructions with reflected minimal surfaces, a replica-trick/CFT formulation, and Engelhardt-Wall CPT gluing—to establish S_R ≈ 2 E_W at leading order, plus quantum corrections. It verifies the framework in simple quantum models, and connects to the algebraic QFT split property, proposing S_R as a finite regulator for entanglement entropy in QFT via a type-I factor. Together, these results provide a computable, finite, and physically meaningful bridge between quantum information measures and bulk geometric quantities in holography, with implications for continuum QFT and regulator constructions.

Abstract

In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy. From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.

Figures (15)

• Figure 1: When $AB$ is the full CFT Hilbert space, the Gibbs state is canonically purified by a thermofield double state in $\mathcal{H}_{L}\otimes \mathcal{H}_R$, which is dual to a two-sided Schwarzschild black hole geometry. The horizon is the entangling surface for $AB$. For this bipartition $A:B$ of the left CFT, the minimal cross-section of its entanglement wedge is $1/2$ the the area of RT surface (shown in blue) for $AA^\star$, that passes through the horizon. We will generalize this picture to other regions $AB$ and states in holographic theories.
• Figure 2: Some example computations of the reflected minimal surfaces shown here in blue. The left case involves the holographic thermofield double which is dual to the maximal extension of the BTZ black hole. Here $AB$ is the entire $S^1$ boundary region of the left copy of $\mathcal{H}_{CFT}^L$. The right Hilbert space $\mathcal{H}_{CFT}^R$ is naturally drawn on the right hand side of this figure but we choose to place it on top of the other space folding the wormhole along the $AB$ entangling surface, which in this case is the horizon of the black hole. The reflected minimal surfaces, in this case are trivially correct, as can be seen by unfolding. On the right we show the case that was important in the original conjectures of Takayanagi:2017knl where the CFT is in the ground state and is cut into three regions $A$, $B$ and $(AB)^c$. The entanglement wedge in this case is not disconnected and shown in white. We have sketched the doubled space $rr^\star(AB)$ which in this case has the topology of a cylinder - similar to the wormhole slice of the eternal black hole. The reflected minimal surface wraps the horizon of this wormhole.
• Figure 3: Some more examples of reflected minimal surface. On the left we show a case where $(\partial A) \cap (\partial B) \neq 0$ and which is a small deformation of the right panel in Figure \ref{['fig:sr']}. The right figure demonstrates what happens in the presence of a mixed state/black hole. The black hole horizon in this case acts also like a mirror.
• Figure 4: These figures demonstrate the fact that $I(A,B^\star) = 0$. The left figure follows the representation of the thermofield double discussed in Figure \ref{['fig:sr']}. The dominant configuration follows the black dashed curves, while the blue curves (and associated homology region) has a larger area. Only the blue curves pass through the AB entangling surface. When projecting the blue curves onto a single wedge region $r(AB)$ one can cut and join these curves so they form surfaces with the same boundary condition as the $A,B$ minimal surfaces. They thus have larger area. This means the minimal entangling surfaces are always disconnected such that the mutual information vanishes. On the right we have shown the equivalent picture in the non-local case.
• Figure 5: Keeping the doubled entanglement wedge $rr^\star(AB)$ fixed we consider reduced sub-regions $C,D^\star$ and their associated minimal surface. We consider only the case where on the original copy $C\cap D = 0$. We can give the same argument as in Fig \ref{['fig:cr']} to show that $I(C,D^\star) =0$.

Theorems & Definitions (1)

• Conjecture 1