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arXiv:1510.00018 · Journal of Statistical Mechanics Theory and Experiment

Rényi information from entropic effects in one higher dimension

Mohammad F. Maghrebi

Published Apr 1, 2016• 3 citations
arXivHTMLPDF

Abstract

Abstract. Computing entanglement entropy and its cousins is often challenging even in the simplest continuum and lattice models, partly because such entropies depend nontrivially on all geometric characteristics of the entangling region. Quantum information measures between two or more regions are even more complicated, but contain more, and universal, information. In this paper, we focus on Rényi entropy and information of the order n = 2. For a free field theory, we show that these quantities are mapped to the change of the thermodynamic free energy by introducing boundaries subject to Dirichlet and Neumann boundary conditions in one higher dimension. This mapping allows us to exploit the powerful tools available in the context of thermal Casimir effect, specifically a multipole expansion suited for computing the Rényi information between arbitrarily-shaped regions. In particular, we compute the Rényi information between two disk-shaped regions at an arbitrary separation distance. We provide an alternative representation of the Rényi information as a sum over closed-loop polymers, which establishes a connection to purely entropic effects, and proves useful in deriving information inequalities. Finally, we discuss extensions of our results beyond free field theories.

DOI
10.1088/1742-5468/2016/04/043102

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Abstract. Computing entanglement entropy and its cousins is often challenging even in the simplest continuum and lattice models, partly because such entropies depend nontrivially on all geometric characteristics of the entangling region. Quantum information measures between two or more regions are even more complicated, but contain more, and universal, information. In this paper, we focus on Rényi entropy and information of the order n = 2. For a free field theory, we show that these quantities are mapped to the change of the thermodynamic free energy by introducing boundaries subject to Dirichlet and Neumann boundary conditions in one higher dimension. This mapping allows us to exploit the powerful tools available in the context of thermal Casimir effect, specifically a multipole expansion suited for computing the Rényi information between arbitrarily-shaped regions. In particular, we compute the Rényi information between two disk-shaped regions at an arbitrary separation distance. We provide an alternative representation of the Rényi information as a sum over closed-loop polymers, which establishes a connection to purely entropic effects, and proves useful in deriving information inequalities. Finally, we discuss extensions of our results beyond free field theories.
We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in d dimensions, the nth Rényi entropy S_{A}^{(n)} behaves as S_{A}^{(n)}=[f/(2πn)^{d-1}][Area(∂A)/(d-2)ε^{d-2}](1+O(n)) in the n→0 limit when the boundary of the entanglement domain A is spherical with the UV cutoff ε. The theory dependence is encapsulated in the cosmological constant f in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain A. In two dimensions, we can use the hot spot idea, which describes the effective action in the high-temperature limit when the temperature is position-dependent, to derive more powerful formulas valid for arbitrary positive n. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.
We present a number of explicit calculations of Renyi and entanglement entropies in situations where the entangling surface intersects the boundary of d -dimensional Minkowski spacetime. When the boundary is a single plane we compute the contribution to the entropy due to this intersection, first in the case of the Neumann and Dirichlet boundary conditions, and then in the case of a generic Robin type boundary condition. The flow in the boundary coupling between the Neumann and Dirichlet phases is analyzed in arbitrary dimension d and is shown to be monotonic, the peculiarity of d = 3 case is noted. We argue that the translational symmetry along the entangling surface is broken due the presence of the boundary which reveals that the entanglement is not homogeneous. In order to characterize this quantitatively, we introduce a density of entanglement entropy and compute it explicitly. This quantity clearly indicates that the entanglement is maximal near the boundary. We then consider the situation where the boundary is composed of two parallel planes at a finite separation and compute the entanglement entropy as well as its density in this case. The complete contribution to entanglement entropy due to the boundaries is shown not to depend on the distance between the planes and is simply twice the entropy in the case of single plane boundary. Additionally, we find how the area law, the part in the entropy proportional to the area of entire entangling surface, depends on the size of the separation between the two boundaries. The latter is shown to appear in the UV finite part of the entropy.
Universal scaling terms occurring in R\'enyi entanglement entropies have the potential to bring new understanding to quantum critical points in free and interacting systems. Quantitative comparisons between analytical continuum theories and numerical calculations on lattice models play a crucial role in advancing such studies. In this paper, we exactly calculate the universal two-cylinder shape dependence of entanglement entropies for free bosons on finite-size square lattices, and compare to approximate functions derived in the continuum using several different Ans\"atze. Although none of these Ans\"atze are exact in the thermodynamic limit, we find that numerical fits are in good agreement with continuum functions derived using the anti--de Sitter/conformal field theory correspondence, an extensive mutual information model, and a quantum Lifshitz model. We use fits of our lattice data to these functions to calculate universal scalars defined in the thin-cylinder limit, and compare to values previously obtained for the free boson field theory in the continuum.
We consider the mutual Rényi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes R A , B . We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere S d − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C ( n ) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Rényi entropies of a single spherical region for a scalar field theory with a small mass.
Calculations of the entanglement entropy of a spatial region in continuum quantum field theory require boundary conditions on the fields at the fictitious boundary of the region. These boundary conditions impact the treatment of the zero modes of the fields and their contribution to the entanglement entropy. We explore this issue in the simplest example, the c=1 compact-boson conformal field theory in 1+1 dimensions. We consider three different types of boundary conditions: spatial Neumann, temporal Neumann, and Dirichlet. We argue that the first two are well motivated, and show that they lead to the same result for the Renyi entropies as well as the entanglement entropy, including a constant term that corresponds to the Affleck-Ludwig boundary entropy. The last set of boundary conditions is less well motivated, and leads to a different value of the constant term. The two values are related by a duality transformation on the compact boson. We also verify some of our results with heat-kernel methods.
We determine the explicit universal form of the entanglement and Renyi entropies, for regions with arbitrary boundary on a null plane or the light-cone. All the entropies are shown to saturate the strong subadditive inequality. This Renyi Markov property implies that the vacuum behaves like a product state. For the null plane, our analysis applies to general quantum field theories, and we show that the entropies do not depend on the region. For the light-cone, our approach is restricted to conformal field theories. In this case, the construction of the entropies is related to dilaton effective actions in two less dimensions. In particular, the universal logarithmic term in the entanglement entropy arises from a Wess-Zumino anomaly action. We also consider these properties in theories with holographic duals, for which we construct the minimal area surfaces for arbitrary shapes on the light-cone. We recover the Markov property and the universal form of the entropy, and argue that these properties continue to hold upon including stringy and quantum corrections. We end with some remarks on the recently proved entropic a-theorem in four spacetime dimensions.