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

          @misc{ pirsa_PIRSA:10050076,
            doi = {10.48660/10050076},
            url = {https://pirsa.org/10050076},
            author = {Wen, Xiao-Gang},
            keywords = {},
            language = {en},
            title = {Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order},
            publisher = {Perimeter Institute},
            year = {2010},
            month = {may},
            note = {PIRSA:10050076 see, \url{https://pirsa.org}}
          }
          

Abstract

Adiabatic evolutions connect two gapped quantum states in the same phase. We argue that the adiabatic evolutions are closely related to local unitary transformations which define a equivalence relation. So the equivalence classes of the local unitary transformations are the universality classes that define the different phases of quantum system. Since local unitary transformations can remove local entanglements, 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 wave function renormalization, where 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 finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which include all the string-net states.