Abstract
For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups must have depth that is at least linear in the system’s diameter. As a tool for the proof, a manifestly Hamiltonian-independent notion of locally invisible operators is introduced. This gives a sufficient condition for a many-body state not to be generated from a product state by any small depth quantum circuit; this is a many-body entanglement witness.
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Hastings M.B., Wen X.-G.: Quasi-adiabatic continuation of quantum states: The stability of topological ground state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005) arXiv:cond-mat/0503554
Osborne T.J.: Simulating adiabatic evolution of gapped spin systems. Phys. Rev. A 75, 032321 (2007)
Kitaev A., Preskill J.: Topological entanglement entropy. Phys. Rev. Lett. 96, 110404 (2006) arXiv:hep-th/0510092
Levin M., Wen X.-G.: Detecting topological order in a ground state wave function. Phys. Rev. Lett. 96, 110405 (2006)
Zhang Y., Grover T., Turner A., Oshikawa M., Vishwanath A.: Quasiparticle statistics and braiding from ground-state entanglement. Phys. Rev. B 85, 235151 (2012) arXiv:1111.2342
Levin M.A., Wen X.-G.: String-net condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005) arXiv:cond-mat/0404617
Flammia S.T., Hamma A., Hughes T.L., Wen X.-G.: Topological entanglement R ényi entropy and reduced density matrix structure. Phys. Rev. Lett. 103, 261601 (2009)
Depenbrock S., McCulloch I.P., Schollwöck U.: Nature of the spin-liquid ground state of the S = 1/2 Heisenberg model on the kagome lattice. Phys. Rev. Lett. 109, 067201 (2012)
Jiang H.-C., Wang Z., Balents L.: Identifying topological order by entanglement entropy. Nat. Phys. 8, 902–905 (2012) arXiv:1205.4289
Kim I.H.: Perturbative analysis of topological entanglement entropy from conditional independence. Phys. Rev. B 86, 245116 (2012)
Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. Math. 162, 581 (2005) arXiv:math/0203060
Kitaev A.: Anyons in an exactly solved model and beyond. Ann. Phys. 321, 2–111 (2006) arXiv:cond-mat/0506438
Bravyi S., Hastings M.B., Verstraete F.: Lieb-Robinson bounds and the generation of correlations and topological quantum order. Phys. Rev. Lett. 97, 050401 (2006)
Eisert J., Osborne T.J.: General entanglement scaling laws from time evolution. Phys. Rev. Lett. 97, 150404 (2006)
König R., Pastawski F.: Generating topological order: no speedup by dissipation. Phys. Rev. B 90, 045101 (2014) arXiv:1310.1037
Bravyi S., Hastings M., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010) arXiv:1001.0344
Kitaev A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303, 2–30 (2003) arXiv:quant-ph/9707021
Bakalov, B., Kirillov Jr., A.: Lectures on tensor categories and modular functors, University Lecture Series, vol. 21. American Mathematical Society, Providence (2001)
Wick G.C., Wightman A.S., Wigner E.P.: The intrinsic parity of elementary particles. Phys. Rev. 88, 101–105 (1952)
Preskill, J.: Topological quantum computation, Lecture notes for Physics 219, Caltech (2004). http://www.theory.caltech.edu/~preskill/ph219/topological
Bombin H., Martin-Delgado M.A.: A family of non-abelian Kitaev models on a lattice: topological confinement and condensation. Phys. Rev. B 78, 115421 (2008) arXiv:0712.0190
Bravyi, S.B., Kitaev, A.Y.: Quantum codes on a lattice with boundary (1998). arXiv:quant-ph/9811052
Kitaev A., Kong L.: Models for gapped boundaries and domain walls. Commun. Math. Phys. 313, 351–373 (2011) arXiv:1104.5047
Bravyi S., Hastings M.B.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609–627 (2011) arXiv:1001.4363
Michalakis S., Zwolak J.: Stability of frustration-free hamiltonians. Commun. Math. Phys. 322, 277–302 (2013) arXiv:1109.1588
Verlinde E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988)
Fiedler L., Naaijkens P.: Haag duality for kitaev’s quantum double model for abelian groups. Rev. Math. Phys. 27, 1550021 (2015)
Naidu D.: Categorical morita equivalence for group-theoretical categories. Commun. Algebra 35, 3544–3565 (2007) arXiv:math/0605530
Haah J.: Local stabilizer codes in three dimensions without string logical operators. Phys. Rev. A 83, 042330 (2011) arXiv:1101.1962
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Communicated by M. M. Wolf
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Haah, J. An Invariant of Topologically Ordered States Under Local Unitary Transformations. Commun. Math. Phys. 342, 771–801 (2016). https://doi.org/10.1007/s00220-016-2594-y
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DOI: https://doi.org/10.1007/s00220-016-2594-y