Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

TL;DR

This paper explores how local unitary transformations relate to long-range entanglement and topological order, providing a classification scheme and an algorithm for wave function renormalization in gapped quantum systems.

Contribution

It introduces a classification of topological orders based on fixed-point wave functions and develops an algorithm for wave function renormalization using local unitary transformations.

Findings

Classified topological orders via fixed-point wave functions.

Developed an algorithm for tensor-network wave function flow.

Connected local entanglement patterns to topological phases.

Abstract

Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have \emph{finite} dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor-product state.