We provide a derivation of holographic entanglement entropy for spherical
entangling surfaces. Our construction relies on conformally mapping the
boundary CFT to a hyperbolic geometry and observing that the vacuum state is
mapped to a thermal state in the latter geometry. Hence the conformal
transformation maps the entanglement entropy to the thermodynamic entropy of
this thermal state. The AdS/CFT dictionary allows us to calculate this
thermodynamic entropy as the horizon entropy of a certain topological black
hole. In even dimensions, we also demonstrate that the universal contribution
to the entanglement entropy is given by A-type trace anomaly for any CFT,
without reference to holography.Comment: 42 pages, 2 figures, few new ref's and comments adde
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. *
Relative entropy between two states in the same Hilbert space is a fundamental statistical measure of the distance between these states. Relative entropy is always positive and increasing with the system size. Interestingly, for two states which are infinitesimally different to each other, vanishing of relative entropy gives a powerful equation ∆S = ∆H for the first order variation of the entanglement entropy ∆S and the expectation value of the modular Hamiltonian ∆H. We evaluate relative entropy between the vacuum and other states for spherical regions in the AdS/CFT framework. We check that the relevant equations and inequalities hold for a large class of states, giving a strong support to the holographic entropy formula. We elaborate on potential uses of the equation ∆S = ∆H for vacuum state tomography and obtain modified versions of the Bekenstein bound.
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.
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