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Aspects of the reflected entropy

Souvik Dutta

Published Mar 8, 2021• 0 citations
OpenAlex

Abstract

The aim of this thesis is to compile our study of a quantum information quantity, called the reflected entropy, which is a measure of quantum correlations in mixed states. Our work is motivated by, and generalizes, an earlier work on the thermo-field double purification of thermal states, in the purview of the AdS/CFT correspondence. Our main result is that in so-called holographic CFT states, the reflected entropy is given by the area of a class of bulk geometric quantities inside the entanglement wedge, called reflected minimal surfaces. From the bulk point of view, we show that half the area of the reflected minimal surface, in Planck units, gives a reinterpretation of the notion of the entanglement wedge cross-section. In doing so, we address the connection with a recent conjecture on the entanglement of purification. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an Engelhardt-Wall holographic construction of the canonical purification − the reflected minimal surfaces are simply Ryu-Takayanagi surfaces in this new spacetime. When we move to the continuum limit, we find a relation to the split property − the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor. In this limit, we show that the reflected entropy serves as a regulator for the UV divergent entanglement entropy. Finally, we consider the reflected entropy for free fermions in two dimensions. Working directly in the continuum theory, we extract the reflected entropy from the spectrum of a singular integral equation whose kernel is determined by the modular flowed free fermion correlation function. We find the spectrum numerically and analytically in certain limits, in one of which, the reflected entanglement spectrum rapidly approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the vacuum/ground state wave function.

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A bstract In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy . From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.

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The aim of this thesis is to compile our study of a quantum information quantity, called the reflected entropy, which is a measure of quantum correlations in mixed states. Our work is motivated by, and generalizes, an earlier work on the thermo-field double purification of thermal states, in the purview of the AdS/CFT correspondence. Our main result is that in so-called holographic CFT states, the reflected entropy is given by the area of a class of bulk geometric quantities inside the entanglement wedge, called reflected minimal surfaces. From the bulk point of view, we show that half the area of the reflected minimal surface, in Planck units, gives a reinterpretation of the notion of the entanglement wedge cross-section. In doing so, we address the connection with a recent conjecture on the entanglement of purification. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an Engelhardt-Wall holographic construction of the canonical purification − the reflected minimal surfaces are simply Ryu-Takayanagi surfaces in this new spacetime. When we move to the continuum limit, we find a relation to the split property − the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor. In this limit, we show that the reflected entropy serves as a regulator for the UV divergent entanglement entropy. Finally, we consider the reflected entropy for free fermions in two dimensions. Working directly in the continuum theory, we extract the reflected entropy from the spectrum of a singular integral equation whose kernel is determined by the modular flowed free fermion correlation function. We find the spectrum numerically and analytically in certain limits, in one of which, the reflected entanglement spectrum rapidly approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the vacuum/ground state wave function.
A bstract In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy . From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using a recently introduced approach to holographic modular flow. We also consider an explicit holographic construction of the canonical purification, introduced by Engelhardt and Wall; the reflected minimal surfaces are simply RT surfaces in this new spacetime. We contrast our results with the entanglement of purification conjecture, and finally comment on the continuum limit where we find a relation to the split property: the reflected entropy computes the von Neumann entropy of a canonical splitting type-I factor introduced by Doplicher and Longo.
Abstract Recently, the reflected entropy is proposed in holographic approach to describe the entanglement of a bipartite quantum system in a mixed state, which is identified as the area of the reflected minimal surface inside the entanglement wedge. In this paper, we study the reflected entropy in the doubly holographic setup, which contains the degrees of freedom of quantum matter in the bulk. In this context, we propose a notion of quantum entanglement wedge cross-section, which may describe the reflected entropy with higher-order quantum corrections. We numerically compute the reflected entropy in pure AdS background and black hole background in four dimensions, respectively. In general, the reflected entropy contains the contribution from the geometry on the brane and the contribution from the CFT. We compute their proportion for different Newton constants and find that their behaviors are in agreement with the results based on the semi-classical gravity and the correlation of CFT coupled to the bath CFT.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
A bstract We consider the reflected entropy and the associated entanglement spectrum for free fermions reduced to two intervals in 1 + 1 dimensions. Working directly in the continuum theory the reflected entropy can be extracted from the spectrum of a singular integral equation whose kernel is determined by the known free fermion modular evolved correlation function. We find the spectrum numerically and analytically in certain limits. For intervals that almost touch the reflected entanglement spectrum approaches the spectrum of the thermal density matrix. This suggests that the reflected entanglement spectrum is well suited to the task of extracting physical data of the theory directly from the ground state wave function.
A bstract Exploiting the split property of quantum field theories (QFTs), a notion of von Neumann entropy associated to pairs of spatial subregions has been recently proposed both in the holographic context — where it has been argued to be related to the entanglement wedge cross section — and for general QFTs. We argue that the definition of this “reflected entropy” can be canonically generalized in a way which is particularly suitable for orbifold theories — those obtained by restricting the full algebra of operators to those which are neutral under a global symmetry group. This turns out to be given by the full-theory reflected entropy minus an entropy associated to the expectation value of the “twist” operator implementing the symmetry operation. Then we show that the reflected entropy for Gaussian fermion systems can be simply written in terms of correlation functions and we evaluate it numerically for two intervals in the case of a two-dimensional Dirac field as a function of the conformal cross-ratio. Finally, we explain how the aforementioned twist operators can be constructed and we compute the corresponding expectation value and reflected entropy numerically in the case of the ℤ 2 bosonic subalgebra of the Dirac field.
A bstract We derive dynamics of the entanglement wedge cross section from the reflected entropy for local operator quench states in the holographic CFT. By comparing between the reflected entropy and the mutual information in this dynamical setup, we argue that (1) the reflected entropy can diagnose a new perspective of the chaotic nature for given mixed states and (2) it can also characterize classical correlations in the subregion/subregion duality. Moreover, we point out that we must improve the bulk interpretation of a heavy state even in the case of well-studied entanglement entropy. Finally, we show that we can derive the same results from the odd entanglement entropy. The present paper is an extended version of our earlier report arXiv:1907.06646 and includes many new results: non-perturbative quantum correction to the reflected/odd entropy, detailed analysis in both CFT and bulk sides, many technical aspects of replica trick for reflected entropy which turn out to be important for general setup, and explicit forms of multi-point semi- classical conformal blocks under consideration.