Reflected entropy, symmetries and free fermions

Bueno, Pablo; Casini, Horacio

doi:10.1007/jhep05(2020)103

Cited by 45 publications

(57 citation statements)

References 40 publications

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“…This is why the reflected entropy is so useful for us in understanding if/how the split property holds in a given theory for a given split distance, s. For more comprehensive discussion on this point, we direct the reader to Refs. [54,75,77]. The split property has been shown to follow from the nuclearity condition which we describe now.…”

Section: B Review Of the Split Property Of Qft And Geometric Regulatorsmentioning

confidence: 75%

TT¯ , the entanglement wedge cross section, and the breakdown of the split property

Asrat

Kudler-Flam

2020

Phys. Rev. D

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We consider fine-grained probes of the entanglement structure of two-dimensional conformal field theories deformed by the irrelevant double-trace operator TT and its closely related but nonetheless distinct single-trace counterpart. For holographic conformal field theories, these deformations can be interpreted as modifications of bulk physics in the ultraviolet region of anti-de Sitter space. Consequently, we can use the Ryu-Takayanagi formula and its generalizations to mixed state entanglement measures to test highly nontrivial consistency conditions. In general, the agreement between bulk and boundary quantities requires the equivalence of partition functions on manifolds of arbitrary genus. For the single-trace deformation, which is dual to an asymptotically linear dilaton geometry, we find that the mutual information and reflected entropy diverge for disjoint intervals when the separation distance approaches a minimum, finite value that depends solely on the deformation parameter. This implies that the mutual information fails to serve as a geometric regulator which is related to the breakdown of the split property at the inverse Hagedorn temperature. In contrast, for the double-trace deformation, which is dual to anti-de Sitter space with a finite radial cutoff, we find all divergences to disappear including the standard quantum field theory ultraviolet divergence that is generically seen as disjoint intervals become adjacent. We furthermore compute reflected entropy in conformal perturbation theory. While we find formally similar behavior between bulk and boundary computations, we find quantitatively distinct results. We comment on the interpretation of these disagreements and the physics that must be altered to restore consistency. We also briefly discuss the TJ and JT deformations.

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Section: B Review Of the Split Property Of Qft And Geometric Regulatorsmentioning

confidence: 75%

TT¯ , the entanglement wedge cross section, and the breakdown of the split property

Asrat

Kudler-Flam

2020

Phys. Rev. D

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“…Since the cross ratio η ≡ (e −a+t +e b−t )(e b+t +e −a−t ) goes to 0 when t is large, we can use the approximate formula of reflected entropy in free fermion theory [56] S R (ρB L :ρB R ) ∼ c(−0.15η ln η + 0.67η) . Notice that in this case the conformal factors Ω i are cancelled upon normalization as h g A and h g −1 B in (5.16) because for these operators we have h i = n n nh i (n n n = 1) (see (5.17)).…”

Section: Black Hole and Black Holementioning

confidence: 99%

“…Recently a new quantity independent of entanglement entropy, called reflected entropy has been introduced [50](also see [51][52][53][54][55][56] for further development). The reflected entropy quantifies an amount of total correlation, including quantum entanglement, for bipartite mixed states ρ AB acting on H AB = H A ⊗ H B .…”

Section: Introductionmentioning

confidence: 99%

Reflected entropy for an evaporating black hole

Chu

Zhou

2020

J. High Energ. Phys.

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We study reflected entropy as a mixed state correlation measure in black hole evaporation. As a measure for bipartite mixed states, reflected entropy can be computed between black hole and radiation, radiation and radiation, and even black hole and black hole. We compute reflected entropy curves in three different models: 3-side wormhole model, End-of-the-World (EOW) brane model in three dimensions and two-dimensional eternal black hole plus CFT model. For 3-side wormhole model, we find that reflected entropy is dual to island cross section. The reflected entropy between radiation and black hole increases at early time and then decreases to zero, similar to Page curve, but with a later transition time. The reflected entropy between radiation and radiation first increases and then saturates. For the EOW brane model, similar behaviors of reflected entropy are found.We propose a quantum extremal surface for reflected entropy, which we call quantum extremal cross section. In the eternal black hole plus CFT model, we find a generalized formula for reflected entropy with island cross section as its area term by considering the right half as the canonical purification of the left. Interestingly, the reflected entropy curve between the left black hole and the left radiation is nothing but the Page curve. We also find that reflected entropy between the left black hole and the right black hole decreases and goes to zero at late time. The reflected entropy between radiation and radiation increases at early time and saturates at late time.

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“…So far, it has not been rigorously proven that reflected entropy should be finite in general, 3 although it is believed to be so at least for most QFTs -see [9] and also [34][35][36]. This was proven to be the case for free fermions in (1 + 1) dimensions in [9] and confirmed later in [5], where we explicitly evaluated it for that theory as a function of the conformal cross ratio. The calculations in [31] also yield finite answers.…”

Section: Jhep11(2020)148mentioning

confidence: 55%

“…The main purpose of this paper is to continue developing the general technology required for the evaluation of reflected entropy for Gaussian systems. As mentioned above, this was started in our previous paper [5], where we obtained general formulas valid for free fermions in arbitrary dimensions. The focus here will be on free scalars, for which we will provide analogous expressions.…”

Section: Jhep11(2020)148mentioning

confidence: 96%

Reflected entropy for free scalars

Bueno

Casini

2020

J. High Energ. Phys.

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We continue our study of reflected entropy, R(A, B), for Gaussian systems. In this paper we provide general formulas valid for free scalar fields in arbitrary dimensions. Similarly to the fermionic case, the resulting expressions are fully determined in terms of correlators of the fields, making them amenable to lattice calculations. We apply this to the case of a (1 + 1)-dimensional chiral scalar, whose reflected entropy we compute for two intervals as a function of the cross-ratio, comparing it with previous holographic and free-fermion results. For both types of free theories we find that reflected entropy satisfies the conjectural monotonicity property R(A, BC) ≥ R(A, B). Then, we move to (2 + 1) dimensions and evaluate it for square regions for free scalars, fermions and holography, determining the very-far and very-close regimes and comparing them with their mutual information counterparts. In all cases considered, both for (1 + 1)- and (2 + 1)-dimensional theories, we verify that the general inequality relating both quantities, R(A, B) ≥ I (A, B), is satisfied. Our results suggest that for general regions characterized by length-scales LA ∼ LB ∼ L and separated a distance ℓ, the reflected entropy in the large-separation regime (x ≡ L/ℓ ≪ 1) behaves as R(x) ∼ −I(x) log x for general CFTs in arbitrary dimensions.

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Reflected entropy, symmetries and free fermions

Cited by 45 publications

References 40 publications

TT¯ , the entanglement wedge cross section, and the breakdown of the split property

TT¯ , the entanglement wedge cross section, and the breakdown of the split property

Reflected entropy for an evaporating black hole

Reflected entropy for free scalars

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