arXiv subjects

H. Casini

Publications and source records attributed to H. Casini.

At least 19 recordsLinked to original sources

arXiv2020

Reduced density matrix and internal dynamics for multicomponent regions

We find the density matrix corresponding to the vacuum state of a massless Dirac field in two dimensions reduced to a region of the space formed by several disjoint intervals. We calculate explicitly its spectral decomposition. The imaginary powers of the density matrix is a unitary operator implementing an internal time flow (the modular flow). We show that in the case of more than one interval this evolution is non-local, producing both, advance in the causal structure and "teleportation" between the disjoint intervals. However, it only mixes the fields on a finite number of trajectories, one for each interval. As an application of these results we compute the entanglement entropy for the massive multi-interval case in the small mass limit.

hep-th↗

arXiv2012

On the RG running of the entanglement entropy of a circle

We show, using strong subadditivity and Lorentz covariance, that in three dimensional space-time the entanglement entropy of a circle is a concave function. This implies the decrease of the coefficient of the area term and the increase of the constant term in the entropy between the ultraviolet and infrared fixed points. This is in accordance with recent holographic c-theorems and with conjectures about the renormalization group flow of the partition function of a three sphere (F-theorem). The irreversibility of the renormalization group flow in three dimensions would follow from the argument provided there is an intrinsic definition for the constant term in the entropy at fixed points. We discuss the difficulties in generalizing this result for spheres in higher dimensions.

hep-th↗

arXiv2010

Entanglement entropy for the n-sphere

We calculate the entanglement entropy for a sphere and a massless scalar field in any dimensions. The reduced density matrix is expressed in terms of the infinitesimal generator of conformal transformations keeping the sphere fixed. The problem is mapped to the one of a thermal gas in a hyperbolic space and solved by the heat kernel approach. The coefficient of the logarithmic term in the entropy for 2 and 4 spacetime dimensions are in accordance with previous numerical and analytical results. In particular, the four dimensional result, together with the one reported by Solodukhin, gives support to the Ryu-Takayanagi holographic anzats. We also find there is no logarithmic contribution to the entropy for odd space time dimensions.

hep-th↗

arXiv2010

Wedge reflection positivity

We show there is a positivity property for Wightman functions which is analogous to the reflection positivity for the euclidean ones. The role of euclidean time reflections is played here by the wedge reflections, which change the sign of the time and one of the spatial coordinates.

hep-th↗

arXiv2010

Entropy inequalities from reflection positivity

We investigate the question of whether the entropy and the Renyi entropies of the vacuum state reduced to a region of the space can be represented in terms of correlators in quantum field theory. In this case, the positivity relations for the correlators are mapped into inequalities for the entropies. We write them using a real time version of reflection positivity, which can be generalized to general quantum systems. Using this generalization we can prove an infinite sequence of inequalities which are obeyed by the Renyi entropies of integer index. There is one independent inequality involving any number of different subsystems. In quantum field theory the inequalities acquire a simple geometrical form and are consistent with the integer index Renyi entropies being given by vacuum expectation values of twisting operators in the Euclidean formulation. Several possible generalizations and specific examples are analyzed.

quant-ph↗

arXiv2009

Entanglement entropy in free quantum field theory

In this review we first introduce the general methods to calculate the entanglement entropy for free fields, within the Euclidean and the real time formalisms. Then we describe the particular examples which have been worked out explicitly in two, three and more dimensions.

hep-th↗

arXiv2009

Remarks on the entanglement entropy for disconnected regions

Few facts are known about the entanglement entropy for disconnected regions in quantum field theory. We study here the property of extensivity of the mutual information, which holds for free massless fermions in two dimensions. We uncover the structure of the entropy function in the extensive case, and find an interesting connection with the renormalization group irreversibility. The solution is a function on space-time regions which complies with all the known requirements a relativistic entropy function has to satisfy. We show that the holographic ansatz of Ryu and Takayanagi, the free scalar and Dirac fields in dimensions greater than two, and the massive free fields in two dimensions all fail to be exactly extensive, disproving recent conjectures.

hep-th↗

arXiv2008

Entanglement entropy for a Dirac fermion in three dimensions: vertex contribution

In three dimensions there is a logarithmically divergent contribution to the entanglement entropy which is due to the vertices located at the boundary of the region considered. In this work we find the corresponding universal coefficient for a free Dirac field, and extend a previous work in which the scalar case was treated. The problem is equivalent to find the conformal anomaly in three dimensional space where multiplicative boundary conditions for the field are imposed on a plane angular sector. As an intermediate step of the calculation we compute the trace of the Green function of a massive Dirac field in a two dimensional sphere with boundary conditions imposed on a segment of a great circle.

hep-th↗

arXiv2008

Relative entropy and the Bekenstein bound

Elaborating on a previous work by Marolf et al, we relate some exact results in quantum field theory and statistical mechanics to the Bekenstein universal bound on entropy. Specifically, we consider the relative entropy between the vacuum and another state, both reduced to a local region. We propose that, with the adequate interpretation, the positivity of the relative entropy in this case constitutes a well defined statement of the bound in flat space. We show that this version arises naturally from the original derivation of the bound from the generalized second law when quantum effects are taken into account. In this formulation the bound holds automatically, and in particular it does not suffer from the proliferation of the species problem. The results suggest that while the bound is relevant at the classical level, it does not introduce new physical constraints semiclassically.

hep-th↗

arXiv2008

Analytic results on the geometric entropy for free fields

The trace of integer powers of the local density matrix corresponding to the vacuum state reduced to a region V can be formally expressed in terms of a functional integral on a manifold with conical singularities. Recently, some progress has been made in explicitly evaluating this type of integrals for free fields. However, finding the associated geometric entropy remained in general a difficult task involving an analytic continuation in the conical angle. In this paper, we obtain this analytic continuation explicitly exploiting a relation between the functional integral formulas and the Chung-Peschel expressions for the density matrix in terms of correlators. The result is that the entropy is given in terms of a functional integral in flat Euclidean space with a cut on V where a specific boundary condition is imposed. As an example we get the exact entanglement entropies for massive scalar and Dirac free fields in 1+1 dimensions in terms of the solutions of a non linear differential equation of the Painleve V type.

hep-th↗

arXiv2007

Entropy localization and extensivity in the semiclassical black hole evaporation

We aim to quantify the distribution of information in the Hawking radiation and inside the black hole in the semiclassical evaporation process. The structure quantum field theory forces to consider a shared information between two different regions of space-time. Using this tool, we show that the entropy of a thermal gas at the Unruh temperature underestimates the actual amount of (shared) information present in a region of the Rindler space. Then, we analyze the mutual information between the black hole and the late time radiation region. We show that in the semiclassical picture it is not possible to recover the eventual purity of the initial state in the final Hawking radiation through correlations established during the whole evaporation period, no matter the interactions present in the theory. We find extensivity of the entropy as a consequence of a reduction to a two dimensional conformal problem in a simple approximation. However, this seems not to be guaranteed in a full four dimensional calculation. We also find that a large amount of information is contained in a small approximately flat region of space-time near the point where the horizon begins. This gives place to large violations of the entropy bounds. This problem is not eased by backscattering effects and we argue that a breaking of conformal invariance is necessary to delocalize the entropy. Finally, we indicate that the mutual information could lead to a way to understand the Bekenstein-Hawking black hole entropy which does not require a reduction in degrees of freedom in order to regulate the entanglement entropy. On the contrary a large number of field degrees of freedom at high energies giving place to a Hagedorn transition implements a distance cutoff in the mutual information, which may in consequence turn out to be bounded.

hep-th↗

arXiv2007

A finite entanglement entropy and the c-theorem

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a mixed density matrix with non zero entropy. This is usually called entanglement entropy, and it is known to be divergent in quantum field theory (QFT). However, it is possible to define a finite quantity F(A,B) for two given different subsets A and B which measures the degree of entanglement between their respective degrees of freedom. We show that the function F(A,B) is severely constrained by the Poincare symmetry and the mathematical properties of the entropy. In particular, for one component sets in two dimensional conformal field theories its general form is completely determined. Moreover, it allows to prove an alternative entropic version of the c-theorem for 1+1 dimensional QFT. We propose this well defined quantity as the meaningfull entanglement entropy and comment on possible applications in QFT and the black hole evaporation problem.

hep-th↗

arXiv2007

Mutual information challenges entropy bounds

We consider some formulations of the entropy bounds at the semiclassical level. The entropy S(V) localized in a region V is divergent in quantum field theory (QFT). Instead of it we focus on the mutual information I(V,W)=S(V)+S(W)-S(V\cup W) between two different non-intersecting sets V and W. This is a low energy quantity, independent of the regularization scheme. In addition, the mutual information is bounded above by twice the entropy corresponding to the sets involved. Calculations of I(V,W) in QFT show that the entropy in empty space cannot be renormalized to zero, and must be actually very large. We find that this entropy due to the vacuum fluctuations violates the FMW bound in Minkowski space. The mutual information also gives a precise, cutoff independent meaning to the statement that the number of degrees of freedom increases with the volume in QFT. If the holographic bound holds, this points to the essential non locality of the physical cutoff. Violations of the Bousso bound would require conformal theories and large distances. We speculate that the presence of a small cosmological constant might prevent such a violation.

gr-qc↗

arXiv2006

Universal terms for the entanglement entropy in 2+1 dimensions

We show that the entanglement entropy and alpha entropies corresponding to spatial polygonal sets in $(2+1)$ dimensions contain a term which scales logarithmically with the cutoff. Its coefficient is a universal quantity consisting in a sum of contributions from the individual vertices. For a free scalar field this contribution is given by the trace anomaly in a three dimensional space with conical singularities located on the boundary of a plane angular sector. We find its analytic expression as a function of the angle. This is given in terms of the solution of a set of non linear ordinary differential equations. For general free fields, we also find the small-angle limit of the logarithmic coefficient, which is related to the two dimensional entropic c-functions. The calculation involves a reduction to a two dimensional problem, and as a byproduct, we obtain the trace of the Green function for a massive scalar field in a sphere where boundary conditions are specified on a segment of a great circle. This also gives the exact expression for the entropies for a scalar field in a two dimensional de Sitter space.

hep-th↗

arXiv2006

A c-theorem for the entanglement entropy

The combination of the Lorentz symmetry and the strong subadditive property of the entropy leads to a c-theorem for the entanglement entropy in 1+1 dimensions. We present a simple derivation of this theorem and compare the associated c-functions with the Zamolodchikov's ones for the case of free fields. We discuss the various difficulties which obstacle the naive generalizations of the entropic c-theorem to higher dimensions.

cond-mat.stat-mech↗

arXiv2005

Entanglement and alpha entropies for a massive scalar field in two dimensions

We find the analytic expression of the trace of powers of the reduced density matrix on an interval of length L, for a massive boson field in 1+1 dimensions. This is given exactly (except for a non universal factor) in terms of a finite sum of solutions of non linear differential equations of the Painlevé V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the Sine-Gordon model in agreement with a result by Delfino et al. We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c-function interpolates between the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two dimensional entropy functions.

cond-mat.other↗

arXiv2005

Entanglement and alpha entropies for a massive Dirac field in two dimensions

We present some exact results about universal quantities derived from the local density matrix, for a free massive Dirac field in two dimensions. We first find the trace of powers of the density matrix in a novel fashion, which involves the correlators of suitable operators in the sine-Gordon model. These, in turn, can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Equipped with the previous results, we find the leading terms for the entanglement entropy, both for short and long distances, and showing that in the intermediate regime it can be expanded in a series of multiple integrals. The previous results have been checked by direct numerical calculations on the lattice, finding perfect agreement. Finally, we comment on a possible generalization of the entanglement entropy c-theorem to the alpha-entropies.

cond-mat.other↗

arXiv2004

Geometric entropy, area, and strong subadditivity

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a density matrix with non zero entropy. This geometric entropy is believed to be deeply related to the entropy of black holes. Indeed, previous calculations in the context of quantum field theory, where the result is actually ultraviolet divergent, have shown that the geometric entropy is proportional to the area for a very special type of subsets. In this work we show that the area law follows in general from simple considerations based on quantum mechanics and relativity. An essential ingredient of our approach is the strong subadditive property of the quantum mechanical entropy.

hep-th↗