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 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.
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 study the reflected entropy in (1+1)-dimensional Lifshitz field theory whose groundstate is described by a quantum mechanical model. Starting from tripartite Lifshitz groundstates, both critical and gapped, we derive explicit formulas for the Rényi reflected entropies reduced to two adjacent or disjoint intervals, directly in the continuum. We show that the reflected entropy in Lifshitz theory does not satisfy monotonicity, in contrast to what is observed for free relativistic fields. We analytically compute the full reflected entanglement spectrum for two disjoint intervals, finding a discrete set of eigenvalues which is that of a thermal density matrix. Furthermore, we investigate the Markov gap, defined as the difference between reflected entropy and mutual information, and find it to be universal and nonvanishing, signaling irreducible tripartite entanglement in Lifshitz groundstates. We also obtain analytical results for the reflected entropies and the Markov gap in 2 + 1 dimensions. Finally, as a byproduct of our results on reflected entropy, we provide exact formulas for two other entanglement-related quantities, namely the computable cross-norm negativity and the operator entanglement entropy.
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 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 L A ∼ L B ∼ 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.
Given a specific interacting quantum Hamiltonian in a general spatial dimension, can one access its entanglement properties, such as the entanglement entropy corresponding to the ground state wave function? Even though progress has been made in addressing this question for interacting bosons and quantum spins, as yet there exist no corresponding methods for interacting fermions. Here we show that the entanglement structure of interacting fermionic Hamiltonians has a particularly simple form-the interacting reduced density matrix can be written as a sum of operators that describe free fermions. This decomposition allows one to calculate the Renyi entropies for Hamiltonians which can be simulated via determinantal quantum Monte Carlo calculations, while employing the efficient techniques hitherto available only for free fermions. The method presented works for the ground state, as well as for the thermally averaged reduced density matrix.
We review some classic works on ground state entanglement entropy in ( 1 + 1 )-dimensional free scalar field theory. We point out identifications between the methods for the calculation of entanglement entropy and we show how the formalism developed for the discretized theory can be utilized in order to obtain results in the continuous theory. We specify the entanglement spectrum and we calculate the entanglement entropy for the theory defined on an interval of finite length L . Finally, we derive the modular Hamiltonian directly, without using the modular flow, via the continuum limit of the expressions obtained in the discretized theory. In a specific coordinate system, the modular Hamiltonian assumes the form of a free field Hamiltonian on the Rindler wedge. Published by the American Physical Society 2024