Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
By Xie Chen, Z. Gu, X. Wen. 2010
757 citations recorded by the source corpus.
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.
Subfields: Atomic and Molecular Physics, and Optics
View Paper: https://www.semanticscholar.org/paper/08cfa3306646d8c21ac93a95cf2c311e41186554
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In Atomic and Molecular Physics, and Optics
Citations in the corpus
- Pre-thermal phases of matter protected by time-translation symmetry (2016, neutral): “[5] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys.”
- Quantum Information Meets Quantum Matter (2015, background): “tum circuit needs to have a depth which scales at least linearly with system size. On the other hand, the simulability of local unitary evolution by a local unitary quantum circuit was shown in [28]. [6] gives a general discussion of the local unitary equivalence condition, the relation between topo”
- Fracton phases of matter (2020, background): “[84] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order,” Phys.”
- Hand-waving and interpretive dance: an introductory course on tensor networks (2016, background): “Despite this, physically relevant states often possess additional structure not found in arbitrary states, and as such do not exhibit this pathological complexity, allowing them to be efficiently described and studied.”
- Efficient representation of quantum many-body states with deep neural networks (2017, neutral): “In Ref.”
- Topological bands for ultracold atoms (2018, neutral): “Another direction consists of establishing an equivalence between classes of density matrices using local unitary operations (Chen et al., 2010), and deducing from this equivalence the desired classification of topological phases.”
- Chaos, complexity, and random matrices (2017, background): “Quantum complexity has also proven useful in condensed matter physics where topological phases of matter can be classified in terms of the quantum complexity of ground state wavefunctions [52].”
- Local random quantum circuits are approximate polynomial-designs: numerical results (2012, neutral): “[16, 21, 44]).”