Abstract
We study boundary renormalization group flows between boundary conformal field theories in $1+1$ dimensions using methods of quantum information theory. We define an entropic $g$-function for theories with impurities in terms of the relative entanglement entropy, and we prove that this $g$-function decreases along boundary renormalization group flows. This entropic $g$-theorem is valid at zero temperature, and is independent from the $g$-theorem based on the thermal partition function. We also discuss the mutual information in boundary RG flows, and how it encodes the correlations between the impurity and bulk degrees of freedom. Our results provide a quantum-information understanding of (boundary) RG flow as increase of distinguishability between the UV fixed point and the theory along the RG flow.
Highlights
The best understood situation arises in two-dimensional CFTs with conformal boundaries, which led to the development of boundary CFT (BCFT)
We study boundary renormalization group flows between boundary conformal field theories in 1 + 1 dimensions using methods of quantum information theory
We define an entropic g-function for theories with impurities in terms of the relative entanglement entropy, and we prove that this g-function decreases along boundary renormalization group flows
Summary
We will study boundary RG flows using the relative entropy. The relative entropy provides a measure of statistical distance between the states of the system with different boundary conditions, and we will see that it is closely related to boundary entropy. The reason is that the relative entropy distinguishes the different states too much, and this masks the decrease of g under the RG This suggests the correct path towards the g-theorem: vary the states in order to minimize the contribution from the modular Hamiltonian, while keeping fixed the entanglement entropy. We show that by working with states on the null boundary of the causal domain, the contribution from the modular Hamiltonian becomes a constant, and the impurity entropy is given explicitly as (minus) a relative entropy. We use this result to prove the entropic g-theorem
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