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.
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.
We derive dynamics of the entanglement wedge cross section directly from the two-dimensional holographic CFTs with a local operator quench. This derivation is based on the reflected entropy, a correlation measure for mixed states. We further compare these results with the mutual information and ones for integrable systems. This comparison directly suggests the classical correlation plays an important role in chaotic systems, unlike integrable ones. Besides a local operator quench, we study the reflected entropy in heavy primary states and find a breaking of the subsystem ETH. We checked the above results also hold for the odd entanglement entropy, which is another measure for mixed states related to the entanglement wedge cross section.
A bstract We study the holographic duality between the reflected entropy and the entanglement wedge cross section with the first order correction. In the field theory side, we consider the reflected entropy for ρABm ρ AB m , where ρ AB is the reduced density matrix for two intervals in the ground state. The reflected entropy in the 2d holographic conformal field theories is computed perturbatively up to the first order in m − 1 by using the semiclassical conformal block. In the gravity side, we compute the entanglement wedge cross section in the backreacted geometry by cosmic branes with tension T m which are anchored at the AdS boundary. Comparing both results we find a perfect agreement, showing the duality works with the first order correction in m − 1.
Abstract We consider the time evolution of mixed state correlation measures in two-dimensional conformal field theories, such as logarithmic negativity, odd entropy, and reflected entropy, after quantum quenches of various kinds. These correlation measures, in the holographic context, are all associated to the entanglement wedge cross section. We contrast various classes of conformal field theories, both rational and irrational (pure) conformal field theories. First, for rational conformal field theories, whose dynamics can be well described by the quasi-particle picture, we find all four quantities for disjoint intervals to be proportional, regardless of the specific quench protocol. Second, using the light cone boot- strap, we generalize our results to irrational conformal field theories where we find sharp distinctions from the quasi-particle results and striking differences between mutual information and the other measures. The large surplus of logarithmic negativity relative to mutual information forces us to reconsider what mutual information and logarithmic negativity really measure. We interpret these results as a signature of information scrambling and chaos in irrational theories. These CFT results perfectly agree with our gravitational (holographic) calculations. Furthermore, using holography, we are able to generalize the results to outside of the light cone limit. Finally, due to the breakdown of the quasi-particle picture for irrational theories, we appeal to the “line-tension picture,” motivated by random unitary circuits, as a phenomenological description. We observe that random unitary circuits, with local Hilbert space dimension determined by the Cardy formula, have precisely the same entanglement dynamics as irrational (including holographic) conformal field theories.
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.
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.
We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic conformal field theories (CFTs). The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (three-dimensional anti-de Sitter space and planar Bañados-Teitelboim-Zanelli black hole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.