On the mutual information in conformal field theory

Chen, Bin; Chen, Lin; Hao, Peng-Xiang; Long, Jiang

doi:10.1007/jhep06(2017)096

Cited by 25 publications

(68 citation statements)

References 106 publications

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“…the mutual information, I AB for the system A∪B made by two disjoint subsystems A and B, the quantity I AB does not vanish when the distance between A and B goes to infinity. This result is consistent with the fact that also some connected correlation functions do not vanish in the thermodynamic limit [9,16,21], being the mutual information an upper bound for normalized connected correlators [22,24].…”

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confidence: 88%

Long-distance entanglement in Motzkin and Fredkin spin chains

Dell’Anna

2019

SciPost Phys.

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We derive some entanglement properties of the ground states of two classes of quantum spin chains described by the Fredkin model, for half-integer spins, and the Motzkin model, for integer ones. Since the ground states of the two models are known analytically, we can calculate the entanglement entropy, the negativity and the quantum mutual information exactly. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when the separation is sent to infinity, i.e. these systems are not affected by decoherence. This strongly entangled behavior, occurring both for colorful versions of the models (with spin larger than 1/2 or 1, respectively) and for colorless cases (spin 1/2 and 1), is consistent with the violation of the cluster decomposition property. Moreover we show that this behavior involves disjoint segments located both at the edges and in the bulk of the chains. Contents arXiv:1904.05205v3 [cond-mat.stat-mech]The study of nonlocal properties and their consequences on the dynamics in addition to the violation of the area law for the entanglement entropy are certainly, at the present date, a very challenging field of research. The concept of locality plays a crucial role in physical theories, with far reaching consequences, a fundamental one being the cluster decomposition property [1,2]. This property implies that two-point connected correlation functions go to zero when the separation of the points goes to infinity. This is the reason why two systems very far apart, separated by a large distance, behave independently.Another aspect related to correlations is the quantum entanglement. In bipartite systems the von Neumann or entanglement entropy quantifies how the two parts of the whole system in a quantum state are entangled. This quantity measures non-local quantum correlations and has universal properties, like the fact that, for gapped systems in the ground states, it scales with the area of the boundary of the two subsystems [3]. This property is called area law and is valid for systems with shortrange interactions. In other words, if the interactions are short-ranged the information among the constituents of the system propagates with a finite speed involving a surface surrounding the source of the signal, like an electromagnetic impulse propagating with the speed of light. For critical onedimensional short-range systems the area law is violated logarithmically [4]. Quantum spin chains are promising tools for universal quantum computation [5] and the efficiency may be related to the amount of quantum entanglement. Spin systems with entanglement entropy larger than that dictated by the area law can be used for quantum computing even more efficiently, and breaking down the speed of the propagation of the excitations can represent a breakthrough for quantum information processing.Recently, novel quantum spin models have been introduced, with integer [6-8] (Motzkin model) and half-integer [9, 10] spins (Fredkin model), which...

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confidence: 88%

Long-distance entanglement in Motzkin and Fredkin spin chains

Dell’Anna

2019

SciPost Phys.

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“…At the first sight this turns out to be a formidable problem, because even for the simplest two-sphere case the computation in the OPE of the twist operator involves the one-point functions of the primary operators in the conical geometry, which requires the detailed information of the CFT. Therefore it is really surprising to find that the mutual information of two disjoint spheres presents universal behaviors at the first few leading orders [28].…”

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confidence: 99%

Holographic mutual information of two disjoint spheres

Chen

Fan

et al. 2018

J. High Energ. Phys.

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We study quantum corrections to holographic mutual information for two disjoint spheres at a large separation by using the operator product expansion of the twist field. In the large separation limit, the holographic mutual information is vanishing at the semiclassical order, but receive quantum corrections from the fluctuations. We show that the leading contributions from the quantum fluctuations take universal forms as suggested from the boundary CFT. We find the universal behavior for the scalar, the vector, the tensor and the fermionic fields by treating these fields as free fields propagating in the fixed background and by using the 1/n prescription. In particular, for the fields with gauge symmetries, including the massless vector boson and massless graviton, we find that the gauge parts in the propagators play indispensable role in reading the leading order corrections to the bulk mutual information. *

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“…Another interesting application is related to the study of lattice entanglement Hamiltonians and their relation to the Bisognano-Wichmann ones[104,[107][108][109][110][111]. Besides, one can consider higher dimensional boson and fermion theories, trying to adapt the techniques of Refs [112][113][114][115][116],. at least in the small subsystem limit.…”

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Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories

Zhang

Ruggiero

Calabrese

2019

J. High Energ. Phys.

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We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero, P. Calabrese, Phys. Rev. Lett. 122, 141602 (2019)], of which this work is a completion. The trace distance between two reduced density matrices ρ A and σ A is obtained from the moments tr(ρ A −σ A ) n and taking the limit n → 1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length embedded in the low lying eigenstates of a one-dimensional critical system of length L, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a scale invariant universal function of /L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several lowlying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten n-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of /L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.

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On the mutual information in conformal field theory

Cited by 25 publications

References 106 publications

Long-distance entanglement in Motzkin and Fredkin spin chains

Long-distance entanglement in Motzkin and Fredkin spin chains

Holographic mutual information of two disjoint spheres

Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories

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