Area law for random graph states

Collins, Benoı̂t; Nechita, Ion; Życzkowski, Karol

doi:10.1088/1751-8113/46/30/305302

Cited by 13 publications

(13 citation statements)

References 25 publications

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“…The probability distributions on D d we have considered so far do not make any assumptions on the internal structure of the underlying Hilbert space C d . To address this issue, in [CNŻ10,CNŻ13] the authors introduce and study a new family of ensembles of density matrices, called random graph states, which encode the underlying structure of the Hilbert space. We introduce next these distributions, referring the interested reader to [CNŻ10,CNŻ13] for the details.…”

Section: Entanglement Of Random Quantum Statesmentioning

confidence: 99%

“…It was shown in [CNŻ13] that the area law holds exactly for adapted marginals of graph states, where we allow arbitrary dimensions of subsystem. Note that, for a given (boundary) edge {i, j}, we have d i = d j , the common dimension of the maximally entangled state corresponding to the edge {i, j}.…”

Section: Entanglement Of Random Quantum Statesmentioning

confidence: 99%

“…The following theorem is the main result of [CNŻ13], showing that the area law holds for random graph states, with the appropriate definition of the boundary volume. Moreover, one can compute the correction term to the area law, a quantity which depends on the topology of the graph G. We refer the interested reader to [CNŻ13,Sections 5,6] for the definition of the correction term h G,S and the proofs.…”

Section: 3mentioning

confidence: 99%

“…Moreover, one can compute the correction term to the area law, a quantity which depends on the topology of the graph G. We refer the interested reader to [CNŻ13,Sections 5,6] for the definition of the correction term h G,S and the proofs. where |∂S| is the area of the boundary of the partition {S, T } from Definition 8.5 and h Γ,S is a positive constant, depending on the topology of the graph G and on the partition {S, T } (and independent of N ).…”

Section: 3mentioning

confidence: 99%

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Random matrix techniques in quantum information theory

Collins

Nechita

2015

Journal of Mathematical Physics

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Abstract. The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels. Contents

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Section: Entanglement Of Random Quantum Statesmentioning

confidence: 99%

Section: Entanglement Of Random Quantum Statesmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Random matrix techniques in quantum information theory

Collins

Nechita

2015

Journal of Mathematical Physics

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“…Generally, the notion of an area for some graph is not so obvious and one has to specify an appropriate definition. It has been shown that random graph states generally satisfy an area law on average [56]. For the present case, a possible way out can be achieved by probing the geometry of the region in terms of a given test particle at thermal equilibrium [57,58].…”

Section: Self-similar Ramificationsmentioning

confidence: 94%

Entanglement entropy on finitely ramified graphs

Akal¹

2018

Phys. Rev. D

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We compute the entanglement entropy in a composite system separated by a finitely ramified boundary with the structure of a self-similar lattice graph. We derive the entropy as a function of the decimation factor which determines the spectral dimension, the latter being generically different from the topological dimension. For large decimations, the graph becomes increasingly dense, yielding a gain in the entanglement entropy which, in the asymptotically smooth limit, approaches a constant value. Conversely, a small decimation factor decreases the entanglement entropy due to a large number of spectral gaps which regulate the amount of information crossing the boundary. In line with earlier studies, we also comment on similarities with certain holographic formulations. Finally, we calculate the higher order corrections in the entanglement entropy which possess a log-periodic oscillatory behavior.

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Holographic duality from random tensor networks

Hayden

Nezami

et al. 2016

J. High Energ. Phys.

538

954

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Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main results of the article define a more general form of bulk-boundary correspondence which could be useful for extending holography to other spacetimes.

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Area law for random graph states

Cited by 13 publications

References 25 publications

Random matrix techniques in quantum information theory

Random matrix techniques in quantum information theory

Entanglement entropy on finitely ramified graphs

Holographic duality from random tensor networks

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