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

TL;DR

This paper shows that the finite bond dimension in MERA and MPO representations of critical states impose similar cut-offs, establishing a connection between multiscale entanglement renormalization and matrix product operators.

Contribution

It introduces an explicit mapping between MERA isometries and MPO tensors, linking scale-invariant wavefunctions to thermal state approximations.

Findings

Finite bond dimension in MERA and MPO impose similar cut-offs.

Explicit mapping between MERA isometries and MPO tensors.

Connection established between multiscale entanglement and thermal state approximation.

Abstract

The multiscale entanglement renormalization ansatz (MERA) provides a constructive algorithm for realizing wavefunctions that are inherently scale invariant. Unlike conformally invariant partition functions however, the finite bond dimension $\chi$ of the MERA provides a cut-off in the fields that can be realized. In this letter, we demonstrate that this cut-off 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 $\chi$. This is achieved by constructing an explicit mapping between the isometries of a MERA and the local tensors of the MPO.