Relative entropy and the RG flow

Casini, Horacio; Testé, Eduardo; Torroba, Gonzalo

doi:10.1007/jhep03(2017)089

Cited by 61 publications

(105 citation statements)

References 36 publications

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“…If SSA is saturated for a certain choice of configuration, the density matrix is a quantum Markov state, implying again the structure for

ρ_{scriptA scriptC}

. One can moreover show that the density matrix does in fact factorize as described above by also invoking the saturation of α‐Renyi entropies (for arbitrary α) . In turn, this implies that the mutual information between

scriptA

and

scriptC

also vanishes; likewise

{boldI}_{3} false(scriptA : scriptB : scriptC false) = 0

.…”

Section: Discussionmentioning

confidence: 81%

The Holographic Entropy Arrangement

Hubeny

Rangamani

Rota

2019

Fortschritte der Physik

116

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We develop a convenient framework for characterizing multipartite entanglement in composite systems, based on relations between entropies of various subsystems. This continues the program initiated in [1], of using holography to effectively recast the geometric problem into an algebraic one. We prove that, for an arbitrary number of parties, our procedure identifies a finite set of entropic information quantities that we conveniently represent geometrically in the form of an arrangement of hyperplanes. This leads us to define the holographic entropy arrangement, whose algebraic and combinatorial aspects we explore in detail. Using the framework, we derive three new information quantities for four parties, as well as a new infinite family for any number of parties. A natural construct from the arrangement is the holographic entropy polyhedron which captures holographic entropy inequalities describing the physically allowed region of entropy space. We illustrate how to obtain the polyhedron by winnowing down the arrangement through a sieve to pick out candidate sign‐definite information quantities. Comparing the polyhedron with the holographic entropy cone, we find perfect agreement for 4 parties and corroborating evidence for the conjectured 5‐party entropy cone. We work with explicit configurations in arbitrary (time‐dependent) states leading to both simple derivations and an intuitive picture of the entanglement pattern.

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“…If SSA is saturated for a certain choice of configuration, the density matrix is a quantum Markov state, implying again the structure for

ρ_{scriptA scriptC}

scriptA

and

scriptC

also vanishes; likewise

{boldI}_{3} false(scriptA : scriptB : scriptC false) = 0

.…”

Section: Discussionmentioning

confidence: 81%

The Holographic Entropy Arrangement

Hubeny

Rangamani

Rota

2019

Fortschritte der Physik

116

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“…(3.9) where we indicated that contributions from curvatures are suppressed by the short-distance cutoff [17]. In conformal perturbation theory around the UV fixed point we expect, on dimensional grounds,…”

Section: Modular Hamiltonian and Stress-tensormentioning

confidence: 94%

“…Unitarity and causality dictate that the EE depends on geometric properties of the boundary of the entangling region, and is the same for any Cauchy surface inside the causal diamond. As in [9,17], our strategy will be to deform the Cauchy surface towards the light-cone in order to equate the EE with the relative entropy. In more detail, let σ be the vacuum reduced density matrix for the UV fixed point theory…”

Section: Boundary Rg Flows and Entanglement Entropymentioning

confidence: 99%

“…All the dependence on Σ has to come from the state. The key point, discussed in [9,17], is that the state ρ R brings in explicit dependence on Σ. The reason is that, in order to compare both states ρ and σ, we have to map the algebra of operatorsφ λ (x) of T 1 to that of T 0 , φ λ (x), at a given Cauchy surface Σ.…”

Section: Modular Hamiltonian and Stress-tensormentioning

confidence: 99%

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Irreversibility in quantum field theories with boundaries

Casini

Landea

Torroba

2019

J. High Energ. Phys.

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We study conformal field theories with boundaries, and their boundary renormalization group (RG) flows, using methods from quantum information theory. Positivity of the relative entropy, together with unitarity and Lorentz invariance, give rise to bounds that characterize the irreversibility of such flows. This generalizes the recently proved entropic g-theorem to higher dimensions. In 2 + 1 dimensions with a boundary, we prove the entropic b-theorem -the decrease of the two-dimensional Weyl anomaly under boundary RG flows. In higher dimensions, the bound implies that the leading area coefficient of the entanglement entropy induced by the defect decreases along the flow. Our proof unifies these properties, and provides an information-theoretic interpretation in terms of the distinguishability between the short distance and long distance states. Finally, we establish a sum rule for the change in the area term in theories with boundaries, which could have implications for models with localized gravity.

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“…For d = 2, 3 the proofs led to entropic cfunctions, which interpolate monotonically between the corresponding values at the fixed points, but are in general not stationary there [57][58][59], unlike Zamolodchikov's c-function. More recently, entropic reasoning based on tools that go beyond S has led to additional monotonicity statements and other restrictions on renormalization flows [60][61][62].…”

Section: Introductionmentioning

confidence: 99%

Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs

Anastasiou

Araya

Güijosa

et al. 2019

J. High Energ. Phys.

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We derive a general formula for renormalized entanglement entropy in evendimensional CFTs holographically dual to Einstein gravity in one dimension higher. In order to renormalize, we adapt the Kounterterm method to asymptotically locally AdS manifolds with conical singularities. On the gravity side, the computation considers extrinsic counterterms and the use of the replica trickà la Lewkowycz-Maldacena. The boundary counterterm B d is shown to satisfy a key property, in direct analogy to the Euler density: when evaluated on a conically singular manifold, it decomposes into a regular part plus a codimension-2 version of itself located at the conical singularity. The renormalized entropy thus obtained is shown to correspond to the universal part of the holographic entanglement entropy, which for spherical entangling surfaces is proportional to the central charge a that is the subject of the a-theorem. We also review and elucidate various aspects of the Kounterterm approach, including in particular its full compatibility with the Dirichlet condition for the metric at the conformal boundary, that is of standard use in holography.

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Relative entropy and the RG flow

Cited by 61 publications

References 36 publications

The Holographic Entropy Arrangement

The Holographic Entropy Arrangement

Irreversibility in quantum field theories with boundaries

Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs

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