An Equivalence between the Exact Renormalization Group and Continuous Entanglement Renormalization

As a generalization of Wilsonian renormalization, the Exact Renormalization Group (ERG) describes the flow of effective models as a function of scale, the central object being the field theory partition function and its invariance under an RG flow. Entanglement renormalization, on the other hand, is described by an isometric or unitary tensor network which achieves entangled state preparation of lattice or continuum theories through an iterated process of local entangling and dilation quantum channels. Although renormalization at the level of states and partition functions are similar, a precise connection between the two formalisms has not been established. Here we attempt to fill the gap and show that for free field theories one may develop a "Hilbert space picture" of the ERG, giving a direct equivalence of entanglement renormalization and a subclass of ERG flows. This observation opens the door to understanding ERG flows in terms of quantum channels on the Hilbert space constructed from the path integral, and allows one to leverage tools from quantum information theory in understanding renormalization beyond the context of tensor network constructions.