Entanglement with centers

Ma, Chen‐Te

doi:10.1007/jhep01(2016)070

Cited by 22 publications

(35 citation statements)

References 88 publications

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“…In fact, in general, the subalgebra that identifies the subsystem A has a nontrivial center. As a result, the Hilbert space of the system is a direct sum of tensor products [46,47]. In this work we extend the typicality results on the entanglement entropy in the presence of a center.…”

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Typical entanglement entropy in the presence of a center: Page curve and its variance

Bianchi

Donà

2019

Phys. Rev. D

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In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability P (SA)dSA that a subsystem has entanglement entropy SA. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance.In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, ∆SA/ SA 1, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.Introduction.-In a seminal paper [1], Page showed that, when an isolated quantum system is in a random pure state, the average entropy of a subsystem is close to maximal. This result plays a central role in the analysis of the black hole information puzzle [2][3][4][5][6][7][8][9], in the quantum foundations of statistical mechanics [10][11][12][13][14][15][16][17][18][19][20][21][22][23], in quantum information theory [24][25][26][27][28][29][30][31] and in the study of the quantum nature of spacetime geometry [32][33][34][35][36][37]. The entanglement entropy, expressed as a function of the subsystem size, is often called the Page curve ( Fig. 1). In a quantum system in a pure state, a subsystem A generally has a nonzero entropy because of entanglement with the rest of the system. In this paper we address the question: Is the average entanglement entropy of pure states S A also the typical entropy of the subsystem? To illustrate the significance of this question, let us consider for instance the gas in a room held at fixed temperature. The canonical ensemble allows us to compute the average energy of the gas. However, the configuration of molecules in one room is one realization of this ensemble-we are not averaging over rooms. How close to the average is the energy of this realization? In other words, is the average energy typical? In statistical mechanics, we answer this question by computing the variance (∆E) 2 . In the canonical ensemble, we find ∆E/ E 1 and we conclude that the average energy is typical. Here we investigate the typicality of the entanglement entropy S A of a subsystem by studying its average and variance.A well-studied special case of ...

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confidence: 57%

“…In the presence of a center, the Hilbert space of the system decomposes as a direct sum of tensor products [46,47],…”

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confidence: 99%

Typical entanglement entropy in the presence of a center: Page curve and its variance

Bianchi

Donà

2019

Phys. Rev. D

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“…We, however, will ignore this issue for simplicity in the reminder of this review. For discussions and attempts to define entanglement entropy in gauge theories, see, e.g., Aoki et al (2015); Casini et al (2014); Chen et al (2015); Donnelly (2014); Donnelly andWall (2015, 2016); Ghosh et al (2015); Huang (2015); Hung and Wan (2015); Ma (2016); Pretko and Senthil (2016); Radicevic (2014Radicevic ( , 2016; Soni and Trivedi (2016);and Van Acoleyen et al (2016) and a review by Pretko (2018). density matrix ρ A has enough information for H A to reconstruct every correlation function in the subregion A.…”

Section: A Bipartite Entanglementmentioning

confidence: 99%

Entanglement entropy: Holography and renormalization group

2018

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Entanglement entropy plays a variety of roles in quantum field theory, including the connections between quantum states and gravitation through the holographic principle. This article provides a review of entanglement entropy from a mixed viewpoint of field theory and holography. A set of basic methods for the computation is developed and illustrated with simple examples such as free theories and conformal field theories. The structures of the ultraviolet divergences and the universal parts are determined and compared with the holographic descriptions of entanglement entropy. The utility of quantum inequalities of entanglement are discussed and shown to derive the C-theorem that constrains renormalization group flows of quantum field theories in diverse dimensions. 42 4. Higher dimensions 42 C. The entropic c-theorem in (1 + 1) dimensions 43 D. The F -theorem in (2 + 1) dimensions 44 1. A sketch of the proof 44 2. Constraints from the F -theorem 45 3. Large mass expansion 45 4. Stationarity at UV fixed points 47 E. The entropic C-theorems in d ≥ 4 dimensions 48 F. Holographic RG flow 49 1. Domain wall and gapped RG flows 49 2. Topology change in a gapped phase 50 3. Testing the F -theorem in holography 51 G. Supersymmetric Rényi entropies 52 IX. Outlook 53 Acknowledgments 53 A. Real time formalism for fermions 53 1. Fermionic system on lattice 53 2. Free massive fermionic fields 54 References 55 arXiv:1801.10352v3 [hep-th] 25 Dec 2019 2We normalize the ground state as Ψ|Ψ = 1 so that trtot(ρtot) = 1.

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“…(4.43) with I E A ⊗ G † E A from the lefthand side, we obtain 10 . For the concrete examples of these central operators in gauge theories, please see [35,36,37,38,39]. We will see in the next subsection that the central boundary operators in the reconstructed algebras on A and on A are dual to these central operators in S bulk (E A ; H bulk )∩ S bulk (E A ; H bulk ) in the bulk theories.…”

Section: Emergence Of Central Operatorsmentioning

confidence: 95%

Shrinking of operators in quantum error correction and AdS/CFT

Hirai

2019

J. High Energ. Phys.

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We first show that a class of operators acting on a given bipartite pure state on H A ⊗ H B can shrink its supports on H A ⊗ H B to only H A or H B while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The "subalgebra code with complementary recovery" introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators.

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Entanglement with centers

Cited by 22 publications

References 88 publications

Typical entanglement entropy in the presence of a center: Page curve and its variance

Typical entanglement entropy in the presence of a center: Page curve and its variance

Entanglement entropy: Holography and renormalization group

Shrinking of operators in quantum error correction and AdS/CFT

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