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arXiv:2609.21370

Entanglement entropy in holographic CFTs with generic boundaries

Fabio Ori, Peng-Xiang Hao

hep-thgr-qcquant-phSubmitted Sep 18, 2026• Announced Sep 21, 2026• 0 citations
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Abstract

Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.

Comments
7 + 8 pages, 7 figures

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Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.
Abstract We study the entanglement entropy for time-like subsystem in two-dimensional boundary conformal field theory (BCFT) both from the field theory and holographic point of view. In field theory, we compute the time-like entanglement entropy of a pure time-like interval at zero and finite temperature using the replica technique and analytical continuation. We find that, similar to the ordinary space-like entanglement entropy in BCFT, the time-like entropy also has a bulk phase and a boundary phase which corresponds respectively to the dominance of the identity block in the bulk and boundary OPE channels. However, we find that in Lorentzian BCFT, the time-like entanglement entropy posses a third Regge phase which arises in the Regge limit of the interval, when one endpoint of the time interval approaches the light cone of the mirror image of the other endpoint. We determine the phase diagram for the time-like entanglement entropy. We find that while the time-like entropy is complex in the bulk phase and has a boundary term in the boundary phase, there is no boundary entropy in the Regge phase. Moreover, it can be real or complex depending on which side the Regge limit is approached from. On the gravity side, we obtain the holographic time-like entanglement entropy from the corresponding bulk dual geometries and find exact agreement with the field theory results. The time-like entanglement entropy may be useful to describe the entanglement of a quantum dot on a half line.
We introduce a new quantity, called pseudo-entropy, as a generalization of entanglement entropy via postselection. We expect this quantity to provide a new class of order parameters in quantum many-body systems. In the anti--de Sitter space (AdS)/conformal field theory (CFT) correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We call this geometric computation of pseudo-entropy via the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT the holographic pseudo-entropy. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-selection is performed. We also present properties of the pseudo-entropy for random states. We then calculate the pseudo-entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo-entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo-entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo-entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo-entropy and give its gravity dual.
The Ryu-Takayanagi conjecture provides a holographic description for the entanglement entropy for the strongly coupled holographic CFTs in the semi-classical limit. It proposes that the entanglement entropy is given by the area of the minimal homologous surface in the dual bulk. We show that the common terms of the entanglement entropy for the free massless fermions or bosons on a torus in the high-temperature expansion can be described by the sum of the signed area of extremal surfaces in the BTZ spacetime. The resulting EE and the corresponding bunch of the exgremal surfaces have preferable properties rather than those from the Ryu-Takayanagi conjecture.
This thesis is devoted to a possible generalization of the entanglement entropy and its implication to the holography, especially AdS/CFT correspondence.The AdS/CFT correspondence is a way to define the quantum gravity in d + 1-dimensional asymptotically anti-de Sitter (AdS) spacetime by using the conformal field theory (CFT) in d-dimension.In this context, it is still unclear how we can realize the d + 1-dimensional full bulk geometries from the d-dimensional quantum field theories.The main object of this thesis is the minimal surface not necessarily anchored on the boundary, which has not been derived from the field theory analysis.First, we review how we can obtain minimal surfaces anchored on the boundary from the entanglement entropy.Then, we introduce a generalization of the entanglement entropy for mixed states and discuss its basic properties.Finally, we argue that we can extract the aforementioned minimal surfaces ending on the bulk from the generalized version of the entanglement entropy.Our explicit derivations include the geometries in three-dimensional AdS and BTZ black holes.First and foremost, I would like to express my deepest gratitude to Norihiro Iizuka, my supervisor.I could not reach here without his patient guidance and encouragement.
This thesis is dedicated to the analysis of correlation structure in two quite different ex-amples. In the first one we investigate the entanglement entropy (EE) through conformal inter-faces in two-dimensional conformal field theories. The EE is a measure for the strength of entanglement between two subsystems of a system in a pure state. Interfaces are one-dimensional objects on a two-dimensional surface along which two conformal field theories are glued together. They are called conformal if their gluing conditions is invariant under conformal transformations that preserve the shape of the interface. Special interfaces are topological ones. They are called topological because the partition function does not change when the interface is continuously deformed. We in particular show that in a vast class of conformal field theories the presence of a topological interface at the boundary between two subregions makes the EE get dressed with an additional sub-leading but universal contribution that solely depends on the interface data. This sub-leading contribution can be interpreted as a relative entropy measuring the relative loss of entanglement compared to no interface insertion. We also compute the EE through general conformal interfaces in the critical Ising model and find that its leading order is affected by the interfaces if and only if it is not topological. Any conformal defect can be characterized by its transmissivity. As physically expected, the EE though conformal interfaces in the Ising model decreases for lower transmissivities. The second example is what we call classical holographic codes. They are classical probabilistic codes defined by a network on a uniform tiling of a constant time slice of AdS 3 -spacetime. They share some remarkable properties with certain quantum error correcting codes that are designed to mimic particular properties of the AdS/CFT correspondence. Under these features are the Ryu-Takayanagi formula and bulk reconstruction properties that both reflect deep connections between the correlation structure of a theory and the ge- ometry of its dual description. Our classical codes can be seen as toy models for holography that show that the latter features do not necessarily originate from a quantum description.
Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator TT[over ¯]. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough et al. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the TT[over ¯] deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the TT[over ¯]-deformed CFT is the holographic dual of a finite region of spacetime.