Entanglement compression in scale space: From the multiscale entanglement renormalization ansatz to matrix product operators

Abstract

The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wave functions that are inherently scale invariant. Unlike conformally invariant partition functions, however, the finite bond dimension χ of the MERA provides a cutoff in the fields that can be realized. In this paper, we demonstrate that this cutoff is equivalent to the one obtained when approximating a thermal state of a critical Hamiltonian with a matrix product operator (MPO) of finite bond dimension χ . This is achieved by constructing an explicit mapping between the isometries of the MERA and the local tensors of the MPO. In terms of energy scales, our results show that a finite bond dimension MERA is equivalent to introducing both an infrared and an ultraviolet scale, characterizing relevant and irrelevant perturbations on the underlying conformal field theory.

Matrix product states and projected entangled pair states: Concepts, symmetries, theorems

“…Similarly, it is possible to construct MERAs with nontrivial MPO symmetries, and hence allow for the description of critical phases with exact topological symmetries (Bridgeman and . Those two facts turn out to be intimately related to each other, as the entanglement Hamiltonian of a MERA is of the exact MPO form (Van Acoleyen et al, 2020).…”

“…The construction of MERAs (and TTNs) can be associated with a coarse tessellation of an antide Sitter geometry, where the renormalization direction coincides with the radial coordinate (Evenbly and Vidal, 2011;Swingle, 2012). The 1D MERA construction can be interpreted as a quantum circuit that implements a conformal mapping between the physical Hilbert space and the renormalized one in scale space (Czech et al, 2016); the entanglement spectrum of the MERA can be identified with that of a MPO representing a thermal state, hence relating the bond dimension of the MERA approximation to the bond dimension needed to represent thermal states using MPOs (Van Acoleyen et al, 2020). For MERAs in 1D it is straightforward to show that they display a logarithmic correction to the area law associated with CFTs by simply finding the shortest path in the MERA embracing the region in which one is interested (Vidal, 2008) [on the other hand, 2D MERAs can be embedded in PEPSs and thus obey an area law (Barthel, Kliesch, and Eisert, 2010)].…”

Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems

“…First of all, the construction of MERAs (and TTNs) can be associated to a coarse tesselation of an Anti-de Sitter geometry, where the renormalization direction coincides with the radial coordinate (Evenbly and Vidal, 2011;Swingle, 2012). The 1-D MERA construction can be interpreted as a quantum circuit which implements a conformal mapping between the physical Hilbert space and the (renormalized) one in scale space (Czech et al, 2016); the entanglement spectrum of the MERA can be identified with the one of a MPO representing a thermal state, hence relating the bond dimension of the MERA approximation to the bond dimension needed to represent thermal states using MPOs (Van Acoleyen et al, 2020). For MERA in 1-D it is straightforward to show that they display the logarithmic correction to the area law associated to CFTs by simply finding the shortest path in the MERA embracing the region in which one is interested .…”

“…Similarly, it is possible to construct MERA with non-trivial MPO symmetries, and hence allows for the description of critical phases with exact topological symmetries (Bridgeman and Williamson, 2017). Those two facts turn out to be intimitaly related to each other, as the entanglement Hamiltonian of a MERA is exactly of the MPO form (Van Acoleyen et al, 2020).…”