The boundary dual of the bulk symplectic form

Belin, Alexandre; Lewkowycz, Aitor; Sárosi, Gábor

doi:10.1016/j.physletb.2018.10.071

Cited by 78 publications

(107 citation statements)

References 25 publications

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“…We will use φ to denote the collection of bulk fields dual to boundary operators. It was shown in [50] that, at leading order in large N , the Berry curvature for normalised states of the above form matches exactly with the symplectic form of the bulk fields, where bulk field variations δφ are related with changes in boundary sources δλ via the holographic dictionary.…”

Section: Holographic Uhlmann Holonomymentioning

confidence: 69%

“…The parameters λ are the coefficients of these operator insertions, and set the boundary conditions for the bulk fields; in the classical limit there is a 1-to-1 map between the boundary conditions λ and bulk field configurations φ. It was shown in [50,51] that the bulk symplectic form is equal to the pullback of the boundary Berry curvature through this map. In light of subregion duality, an immediate question presents itself: is there a generalisation of this result to subregions?…”

Section: Figure 12mentioning

confidence: 99%

“…One sees that λ| is related to |λ by a complex conjugation and reflection of the sources across τ = 0. Following [50], we will refer to this transformation as Z 2 + C, where Z 2 refers to the time relection, and C refers to the complex conjugation. Let us call the reflected manifold on which this state is prepared M + .…”

Section: Holographic Uhlmann Holonomymentioning

confidence: 99%

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The holographic dual of the entanglement wedge symplectic form

Kirklin

2020

J. High Energ. Phys.

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In this paper, we find the boundary dual of the symplectic form for the bulk fields in any entanglement wedge. The key ingredient is Uhlmann holonomy, which is a notion of parallel transport of purifications of density matrices based on a maximisation of transition probabilities. Using a replica trick, we compute this holonomy for curves of reduced states in boundary subregions of holographic QFTs at large N , subject to changes of operator insertions on the boundary. It is shown that the Berry phase along Uhlmann parallel paths may be written as the integral of an abelian connection whose curvature is the symplectic form of the entanglement wedge. This generalises previous work on holographic Berry curvature.

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Section: Holographic Uhlmann Holonomymentioning

confidence: 69%

Section: Figure 12mentioning

confidence: 99%

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The holographic dual of the entanglement wedge symplectic form

Kirklin

2020

J. High Energ. Phys.

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“…First, a proof of the holographic complexity proposals should occur through the equality of bulk and boundary Euclidean partition functions, and the interface between the two sides is governed by the boundary (QFT) sources. While there have already been several attempts in this general direction [45][46][47][48], the present paper is perhaps the first to construct genuine cost functions for a particular class of source configurations. Second, in the path integral optimization program [35][36][37], one optimizes the Liouville action alone, i.e., without including higher order corrections in .…”

Section: Discussion and Outlookmentioning

confidence: 99%

Path Integral Optimization as Circuit Complexity

et al. 2019

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Early efforts to understand complexity in field theory have primarily employed a geometric approach based on the concept of circuit complexity in quantum information theory. In a parallel vein, it has been proposed that certain deformations of the Euclidean path integral that prepares a given operator or state may provide an alternative definition, whose connection to the standard notion of complexity is less apparent. In this letter, we bridge the gap between these two proposals in two-dimensional conformal field theories, by explicitly showing how the latter approach from path integral optimization may be given by a concrete realization within the standard gate counting framework. In particular, we show that when the background geometry is deformed by a Weyl rescaling, a judicious gate counting allows one to recover the Liouville action as a particular choice within a more general class of cost functions.

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“…Alternatively quantum information metric has been studied as a field-theoretic dual of the maximal volume[8][9][10].…”

mentioning

confidence: 99%

Does boundary distinguish complexities?

Sato

Watanabe

2019

J. High Energ. Phys.

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Recently, Chapman et al. argued that holographic complexities for defects distinguish action from volume. Motivated by their work, we study complexity of quantum states in conformal field theory with boundary. In generic two-dimensional BCFT, we work on the path-integral optimization which gives one of field-theoretic definitions for the complexity. We also perform holographic computations of the complexity in Takayanagi's AdS/BCFT model following by the "complexity = volume" conjecture and "complexity = action" conjecture. We find that increments of the complexity due to the boundary show the same divergent structures in these models except for the CA complexity in the AdS 3 /BCFT 2 model as the argument by Chapman et al. Thus, we conclude that boundary does not distinguish the complexities in general.

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The boundary dual of the bulk symplectic form

Cited by 78 publications

References 25 publications

The holographic dual of the entanglement wedge symplectic form

The holographic dual of the entanglement wedge symplectic form

Path Integral Optimization as Circuit Complexity

Does boundary distinguish complexities?

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