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.
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.
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 We discuss two methods that, through a combination of cyclically gluing copies of a given n -party boundary state in AdS/CFT and a canonical purification, creates a bulk geometry that contains a boundary homologous minimal surface with area equal to 2 or 4 times the n -party entanglement wedge cross-section, depending on the parity of the party number and choice of method. The areas of the minimal surfaces are each dual to entanglement entropies that we define to be candidates for the n -party reflected entropy. In the context of AdS 3 /CFT 2 , we provide a boundary interpretation of our construction as a multiboundary wormhole, and conjecture that this interpretation generalizes to higher dimensions.
We discuss two methods that, through a combination of cyclically gluing copies of a given n-party boundary state in AdS/CFT and a canonical purification, creates a bulk geometry that contains a boundary homologous minimal surface with area equal to 2 or 4 times the n-party entanglement wedge cross-section, depending on the parity of the party number and choice of method. The areas of the minimal surfaces are each dual to entanglement entropies that we define to be candidates for the n-party reflected entropy. In the context of AdS3/CFT2, we provide a boundary interpretation of our construction as a multiboundary wormhole, and conjecture that this interpretation generalizes to higher dimensions.
A bstract We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well- supported conjectures about the entanglement wedge cross section surface EW . If EW is related to either the CFT’s reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at O(GN1). O 1 G N . We prove that this implies holographic CFT states must have O(GN1). O 1 G N . amounts of tripartite entanglement. This proof involves a new Fannes-type inequality for the reflected entropy, which itself has many interesting applications.
A bstract In the AdS/CFT context, the entanglement of purification (EoP, denoted as E P ) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy S R is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown E P ≥ 21SR(n) 1 2 S R n , n ≥ 2 for random tensor networks. In this paper, we study this inequality beyond n ≥ 2, and we focus on the range 0 < n < 2. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in [1] to the Rényi case. And we have also presented two methods for Rényi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for 0 < n < 2, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the Rényi Markov gap and the monotonicity of the Rényi reflected entropy in the free scalar theory.
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.