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arXiv:2608.12245

A sharp bound on spacetime distance from quantum entanglement

Zhi-Wei Wang, Arshid Shabir, Mir Faizal, Samuel L. Braunstein

hep-thgr-qcquant-phSubmitted Aug 12, 2026• Announced Aug 13, 2026• 0 citations
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Abstract

Ryu-Takayanagi established how boundary entanglement encodes bulk area. We provide the metric counterpart: boundary mutual information imposes a rigorous lower bound on bulk geodesic separation that diverges logarithmically as correlations vanish. A multiscale iteration promotes this local inequality to a global obstruction to bulk connectivity. For parallel strips in AdS$_5$/CFT$_4$, the bound necessitates a quantum resolution of the classical mutual-information transition and fixes the asymptotic growth of geodesic distance.

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95 pages, 2 figures

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•Jan 1, 2016•1 citation
Ryu-Takayanagi established how boundary entanglement encodes bulk area. We provide the metric counterpart: boundary mutual information imposes a rigorous lower bound on bulk geodesic separation that diverges logarithmically as correlations vanish. A multiscale iteration promotes this local inequality to a global obstruction to bulk connectivity. For parallel strips in AdS5_55​/CFT4_44​, the bound necessitates a quantum resolution of the classical mutual-information transition and fixes the asymptotic growth of geodesic distance.
Abstract In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.
We define a generalized entanglement measure in the context of AdS3_33​/CFT2_22​. Compared to the ordinary entanglement entropy between spatial degrees of freedom dual to the area of the Ryu-Takayanagi surface, we take into account both entanglement between spatial degrees of freedom as well as between different fields of the boundary CFT. We then calculate this generalized entanglement measure in a thermal state dual to the BTZ black hole in the setting of the D1/D5 system at and close to the orbifold point. We find that the entanglement entropy defined in this way is dual to the length of a geodesic with non-zero winding number. Such geodesics probe the entire bulk geometry, including regions known as entanglement shadows which are not reached by any Ryu-Takayanagi surface. This allows us to describe regions close to the black hole horizon in the one-sided black hole and the wormhole growth in the case of a two-sided black hole from entanglement data in the boundary field theory, leading us to propose that entanglement is in fact enough to reconstruct the full BTZ geometry from boundary data.
A bstract Using recent developments in expressing one-loop partition functions in Euclidean AdS 2 space-times in terms of character integrals, we relate the one-loop effective action for a free field theory in AdS 2 (comprised of a massless scalar field and a massless Majorana fermion field) to the partition function of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) models on the two global AdS 2 boundaries. The equal number of bosonic and fermionic degrees in the field theory guarantee that the one-loop calculation is free of all UV divergences except a logarithmic one consistent with the expected entanglement entropy behaviour in a CQM. Via a thermofield double representation, we compute the entanglement entropy between two copies of the CFT 1 (CQM), each living near one of the two boundaries of global AdS 2 , in a state at global time τ → – ∞. This entanglement entropy is expressed in terms of the logarithm of the regularised length of a closed particle trajectory infinitesimally near the rim of the Euclidean AdS 2 disc. We view this relation between boundary quantum entanglement and a bulk geometrical quantity as the AdS 2 / CFT 1 version of the Ryu-Takayanagi conjecture in our setup. The boundary entanglement entropy is equal to 4 times the thermodynamic entropy read off from the regularised one-loop effective action in AdS 2 . Further, we compute the bulk entanglement entropy associated with black hole horizons in Lorentzian AdS 2 and show that it precisely matches the boundary entanglement entropy.
In 2006, Ryu and Takayanagi (RT) pointed out that (with a suitable cutoff) the entanglement entropy between two complementary regions of an equal-time surface of a d+1-dimensional conformal field theory on the conformal boundary of AdS_{d+2} is, when the AdS radius is appropriately related to the parameters of the CFT, equal to 1/4G times the area of the d-dimensional minimal surface in the AdS bulk which has the junction of those complementary regions as its boundary, where G is the bulk Newton constant. We point out here that the RT-equality implies that, in the quantum theory on the bulk AdS background which is related to the boundary CFT according to Rehren's 1999 algebraic holography theorem, the entanglement entropy between two complementary bulk Rehren wedges is equal to 1/4G times the (suitably cut off) area of their shared ridge. (This follows because of the geometrical fact that, for complementary ball-shaped regions, the RT minimal surface is precisely the shared ridge of the complementary bulk Rehren wedges which correspond, under Rehren's bulk-wedge to boundary double-cone bijection, to the complementary boundary double-cones whose bases are the RT complementary balls.) This is consistent with the Bianchi-Meyers conjecture -- that, in a theory of quantum gravity, the entanglement entropy, S, between the degrees of freedom of a given region with those of its complement is S = A/4G (+ lower order terms) -- but only if the phrase 'degrees of freedom' is replaced by 'matter degrees of freedom'. It also supports related previous arguments of the author -- consistent with the author's 'matter-gravity entanglement hypothesis' -- that the AdS/CFT correspondence is actually only a bijection between just the matter (i.e. non-gravity) sector operators of the bulk and the boundary CFT operators.
In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.
In the context of relating AdS/CFT to quantum information theory, we propose a holographic dual of Fisher information metric for mixed states in the boundary field theory. This amounts to a holographic measure for the distance between two mixed quantum states. For a spherical subregion in the boundary we show that this is related to a particularly regularized volume enclosed by the Ryu-Takayanagi surface. We further argue that the quantum correction to the proposed Fisher information metric is related to the quantum correction to the boundary entanglement entropy. We discuss consequences of this connection.