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Register a new userPerturbative expansions of Renyi relative divergences and holography
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Authors
Tomonori Ugajin
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In this paper, we develop a novel way to perturbatively calculate Renyi relative divergences $D_{gamma}(rho|| sigma) ={rm tr} rho^{gamma} sigma^{1-gamma}$ and related quantities without using replica trick as well as analytic continuation. We explicitly determine the form of the perturbative term at any order by an integral along the modular flow of the unperturbed state. By applying the prescription to a class of reduced density matrices in conformal field theory, we find that the second order term of certain linear combination of the divergences has a holographic expression in terms of bulk symplectic form, which is a one parameter generalization of the statement Fisher information = Bulk canonical energy.
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Quantum Renyi relative entropies provide a one-parameter family of distances between density matrices, which generalizes the relative entropy and the fidelity. We study these measures for renormalization group flows in quantum field theory. We derive explicit expressions in free field theory based on the real time approach. Using monotonicity properties, we obtain new inequalities that need to be satisfied by consistent renormalization group trajectories in field theory. These inequalities play the role of a second law of thermodynamics, in the context of renormalization group flows. Finally, we apply these results to a tractable Kondo model, where we evaluate the Renyi relative entropies explicitly. An outcome of this is that Andersons orthogonality catastrophe can be avoided by working on a Cauchy surface that approaches the light-cone.
Collinear and soft divergences in perturbative quantum gravity are investigated to arbitrary orders in amplitudes for wide-angle scattering, using methods developed for gauge theories. We show that collinear singularities cancel when all such divergent diagrams are summed over, by using the gravitational Ward identity that decouples the unphysical polarizations from the S-matrix. This analysis generalizes a result previously demonstrated in the eikonal approximation. We also confirm that the only virtual graviton corrections that give soft logarithmic divergences are of the ladder and crossed ladder type.
We holographically compute the Renyi relative divergence $D_{alpha} (rho_{+} || rho_{-})$ between two density matrices $rho_{+}, ; rho_{-}$ prepared by path integrals with constant background fields $lambda_{pm}$ coupled to a marginal operator in JT gravity. Our calculation is non perturbative in the difference between two sources $ lambda_{+} -lambda_{-}$. When this difference is large, the bulk geometry becomes a black hole with the maximal temperature allowed by the Renyi index $alpha$. In this limit, we find an analytic expression of the Renyi relative divergence, which is given by the on shell action of the back reacted black hole plus the contribution coming from the discontinuous change of the background field.
In this thesis we investigate some aspects of quantum field theories from a holographic perspective. In the first chapters we examine in detail a one-paremeter family of three-dimensional gauge theories by means of their type IIA gravity duals. We analyse features such as their confinement nature, spectrum, entanglement properties or thermal phase transitions. This family interpolates between quasi-conformal and quasi-confining physics. In the last two chapters, we use bottom-up models to study complex conformal field theories and transport properties of dense QCD respectively.
For $alpha,z>0$ with $alpha e1$, motivated by comparison between different kinds of Renyi divergences in quantum information, we consider log-majorization between the matrix functions begin{align*} P_alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^alpha B^{1/2}, Q_{alpha,z}(A,B)&:=(B^{1-alphaover2z}A^{alphaover z}B^{1-alphaover2z})^z end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $alpha,z$ for which $P_alpha(A,B)prec_{log}Q_{alpha,z}(A,B)$ and $Q_{alpha,z}(A,B)prec_{log}P_alpha(A,B)$ holds, respectively.
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