Relative entanglement entropy for widely separated regions in curved spacetime

Hollands, Stefan; Islam, Onirban; Sanders, Ko

doi:10.1063/1.5017093

Cited by 6 publications

(12 citation statements)

References 39 publications

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“…6 applies and provides upper bounds on the entanglement entropy E R pω 0 q of two diamonds of size r in Minkowski space separated by a distance R (or more generally, two bounded regions A and B in a static time-slice C separated by a distance R). One can again get better bounds for large R using techniques from modular theory in a similar way as for scalar fields [82].…”

Section: Free Dirac Fieldsmentioning

confidence: 99%

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Entanglement Measures and Their Properties in Quantum Field Theory

Hollands

Sanders

2018

SpringerBriefs in Mathematical Physics

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An entanglement measure for a bipartite quantum system is a state functional that vanishes on separable states and that does not increase under separable (local) operations. For pure states, essentially all entanglement measures are equal to the v. Neumann entropy of the reduced state, but for mixed states, this uniqueness is lost. In quantum field theory, bipartite systems are associated with causally disjoint regions. But if these regions touch each other, there are no separable normal states to begin with, and one must hence leave a finite "safety-corridor" between the regions. Due to this corridor, the normal states of bipartite systems are necessarily mixed, so the v. Neumann entropy is not a good entanglement measure any more in this sense. In this volume, we study various good entanglement measures. In particular, we study the relative entanglement entropy, E R , defined as the minimum relative entropy between the given state and an arbitrary separable state. We establish upper and lower bounds on this quantity in several situations: 1) In arbitrary CFTs in d `1 dimensions, we provide an upper bound for the entanglement measure of the vacuum state if the two regions of the bipartite system are a diamond and the complement of another diamond. The bound is given in terms of the spins, dimensions of the CFT and the geometric invariants associated with the regions. 2) In integrable models in 1 `1 dimensions defined by a general analytic, crossing symmetric 2-body scattering matrix, we give an upper bound for the entanglement measure of the vacuum state for a pair of diamonds that are far apart, showing exponential decay with the distance between the diamonds. The class of models includes e.g. the Sinh-Gordon field theory. 3) We give upper bounds for our entanglement measure for a free Klein-Gordon/Dirac field in the ground state on an arbitrary static spacetime. Our upper bounds show exponential decay of the entanglement measure for large geodesic distance and an "area law" for small distances (modified by a logarithm). 4) We show that if we add charged particles to an arbitrary state, then E R decreases by a positive amount which is no more than the logarithm of the quantum dimension of the charges (this dimension need not be integer). 5) We establish a lower bound on our entanglement measure for arbitrary regions that get close to each other. This lower bound is of the type of an "area law" with the proportionality constant given by the number N of free fields in the UV-fixed point times a quantity D 2 that can be interpreted as the distillable entanglement of one "Cbit-pair" in the state.

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Section: Free Dirac Fieldsmentioning

confidence: 99%

“…where F is the mean over the group K " OpNq already defined above in (82). dimpλq is by the general theory equal to the statistical dimension of the charged state ρ˚ω 0 .…”

Section: (265)mentioning

confidence: 99%

Entanglement Measures and Their Properties in Quantum Field Theory

Hollands

Sanders

2018

SpringerBriefs in Mathematical Physics

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“…First quantization map. We say that C ⊂ R d−1 is a spatially complete region iff it is open, regular 6 and with regular boundary. 7 Here we work with this kind of regions.…”

Section: Local Algebras At a Fixed Timementioning

confidence: 99%

“…which is much better behaved than the entropy (see for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14] for actual calculations). In fact, the relative entropy has an expression directly in the continuous theory for type III algebras in terms of the Araki formula [15].…”

Section: Introductionmentioning

confidence: 99%

Relative entropy for coherent states from Araki formula

2019

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We make a rigorous computation of the relative entropy between the vacuum state and a coherent state for a free scalar in the framework of algebraic quantum field theory (AQFT). We study the case of the Rindler wedge. Previous calculations including path integral methods and computations from the lattice give a result for such relative entropy which involves integrals of expectation values of the energy-momentum stress tensor along the considered region. However, the stress tensor is in general non-unique. That means that if we start with some stress tensor, then we can "improve" it adding a conserved term without modifying the Poincaré charges. On the other hand, the presence of such an improving term affects the naive expectation for the relative entropy by a non-vanishing boundary contribution along the entangling surface. In other words, this means that there is an ambiguity in the usual formula for the relative entropy coming from the non-uniqueness of the stress tensor. The main motivation of this work is to solve this puzzle. We first show that all choices of stress tensor except the canonical one are not allowed by positivity and monotonicity of the relative entropy. Then we fully compute the relative entropy between the vacuum and a coherent state in the framework of AQFT using the Araki formula and the techniques of modular theory. After all, both results coincide and give the usual expression for the relative entropy calculated with the canonical stress

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“…In particular, we focus on the relative entropy of entanglement E R (ρ AB ) [24,25] among entanglement measures, motivated by recent progresses of computational techniques in CFTs of relative entropies [26,27,28,29,30,31]. Several bounds for REE in quantum field theories have been obtained in [32,33] via an algebraic quantum field theory approach 5 (refer to [35] for an excellent review).…”

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confidence: 99%

Towards an entanglement measure for mixed states in CFTs based on relative entropy

Takayanagi

Ugajin

Umemoto

2018

J. High Energ. Phys.

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Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S(ρ AB ||σ AB ) between a given mixed state ρ AB and an arbitrary separable state σ AB . The REE is always bounded by the mutual information I AB = S(ρ AB ||ρ A ⊗ ρ B ) because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix ρ 0 AB on disjoint subsystems A ∪ B and arbitrarily separable state σ AB in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.

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Relative entanglement entropy for widely separated regions in curved spacetime

Cited by 6 publications

References 39 publications

Entanglement Measures and Their Properties in Quantum Field Theory

Entanglement Measures and Their Properties in Quantum Field Theory

Relative entropy for coherent states from Araki formula

Towards an entanglement measure for mixed states in CFTs based on relative entropy

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