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Scarpa G, Molnár A, Ge Y, García-Ripoll JJ, Schuch N, Pérez-García D, Iblisdir S. Projected Entangled Pair States: Fundamental Analytical and Numerical Limitations. Phys Rev Lett 2020; 125:210504. [PMID: 33275001 DOI: 10.1103/physrevlett.125.210504] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2020] [Accepted: 09/16/2020] [Indexed: 06/12/2023]
Abstract
Matrix product states and projected entangled pair states (PEPS) are powerful analytical and numerical tools to assess quantum many-body systems in one and higher dimensions, respectively. While matrix product states are comprehensively understood, in PEPS fundamental questions, relevant analytically as well as numerically, remain open, such as how to encode symmetries in full generality, or how to stabilize numerical methods using canonical forms. Here, we show that these key problems, as well as a number of related questions, are algorithmically undecidable, that is, they cannot be fully resolved in a systematic way. Our work thereby exposes fundamental limitations to a full and unbiased understanding of quantum many-body systems using PEPS.
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Affiliation(s)
- G Scarpa
- Departamento Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas, Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain
- Universidad Politécnica de Madrid, Escuela Técnica Superior de Ingenieria de Sistemas Informáticos, C/ Alan Turing s/n, 28031 Madrid, Spain
| | - A Molnár
- Departamento Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas, Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain
- Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany
| | - Y Ge
- Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany
| | - J J García-Ripoll
- Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, 28006 Madrid, Spain
| | - N Schuch
- Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 München, Germany
- University of Vienna, Department of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
- University of Vienna, Department of Physics, Boltzmanngasse 5, 1090 Wien, Austria
| | - D Pérez-García
- Departamento Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas, Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain
| | - S Iblisdir
- Departamento Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
- Instituto de Ciencias Matemáticas, Campus Cantoblanco UAM, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain
- Departament de Fsica Quntica i Astronomia & Institut de Cincies del Cosmos, Universitat de Barcelona, 08028 Barcelona, Spain
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Haegeman J, Zauner V, Schuch N, Verstraete F. Shadows of anyons and the entanglement structure of topological phases. Nat Commun 2015; 6:8284. [PMID: 26440783 PMCID: PMC4600714 DOI: 10.1038/ncomms9284] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2015] [Accepted: 08/07/2015] [Indexed: 11/09/2022] Open
Abstract
The low-temperature dynamics of quantum systems are dominated by the low-energy eigenstates. For two-dimensional systems in particular, exotic phenomena such as topological order and anyon excitations can emerge. While a complete low-energy description of strongly correlated systems is hard to obtain, essential information about the elementary excitations is encoded in the eigenvalue structure of the quantum transfer matrix. Here we study the transfer matrix of topological quantum systems using the tensor network formalism and demonstrate that topological quantum order requires a particular type of 'symmetry breaking' for the fixed point subspace. We also relate physical anyon excitations to domain-wall excitations at the level of the transfer matrix. This formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental questions related to anyon condensation and confinement.
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Affiliation(s)
- J Haegeman
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281 S9, B-9000 Ghent, Belgium
| | - V Zauner
- Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria
| | - N Schuch
- JARA Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
| | - F Verstraete
- Department of Physics and Astronomy, University of Ghent, Krijgslaan 281 S9, B-9000 Ghent, Belgium.,Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090 Wien, Austria
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Yang S, Lehman L, Poilblanc D, Van Acoleyen K, Verstraete F, Cirac JI, Schuch N. Edge theories in projected entangled pair state models. Phys Rev Lett 2014; 112:036402. [PMID: 24484150 DOI: 10.1103/physrevlett.112.036402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2013] [Indexed: 06/03/2023]
Abstract
We analyze the low energy excitations of spin lattice systems in two dimensions at zero temperature within the framework of projected entangled pair state models. Perturbations in the bulk give rise to physical excitations located at the edge. We identify the corresponding degrees of freedom, give a procedure to derive the edge Hamiltonian, and illustrate that it can exhibit a rich phase diagram. For topological models, the edge Hamiltonian is constrained by the topological order in the bulk, which gives rise to one-dimensional edge models with unconventional properties; for instance, a topologically ordered bulk can protect a ferromagnetic Ising chain at the edge against spontaneous symmetry breaking.
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Affiliation(s)
- S Yang
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
| | - L Lehman
- Institut für Quanteninformation, RWTH Aachen University, D-52056 Aachen, Germany
| | - D Poilblanc
- Laboratoire de Physique Théorique, C.N.R.S. and Université de Toulouse, F-31062 Toulouse, France
| | - K Van Acoleyen
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium
| | - F Verstraete
- Department of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, 9000 Gent, Belgium and Vienna Center for Quantum Science and Technology, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria
| | - J I Cirac
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany
| | - N Schuch
- Institut für Quanteninformation, RWTH Aachen University, D-52056 Aachen, Germany
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Wahl TB, Tu HH, Schuch N, Cirac JI. Projected entangled-pair states can describe chiral topological states. Phys Rev Lett 2013; 111:236805. [PMID: 24476298 DOI: 10.1103/physrevlett.111.236805] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2013] [Indexed: 06/03/2023]
Abstract
We show that projected entangled-pair states (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a nontrivial Chern number. They are ground states of two different kinds of free-fermion Hamiltonians: (i) local and gapless; (ii) gapped, but with hopping amplitudes that decay according to a power law. We derive general conditions on topological free-fermionic projected entangled-pair states that show that they cannot correspond to exact ground states of gapped, local parent Hamiltonians and provide numerical evidence demonstrating that they can nevertheless approximate well the physical properties of topological insulators with local Hamiltonians at arbitrary temperatures.
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Affiliation(s)
- T B Wahl
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
| | - H-H Tu
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
| | - N Schuch
- Institut für Quanteninformation, RWTH Aachen, D-52056 Aachen, Germany
| | - J I Cirac
- Max-Planck Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany
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Abstract
Superselection rules severely alter the possible operations that can be implemented on a distributed quantum system. Whereas the restriction to local operations imposed by a bipartite setting gives rise to the notion of entanglement as a nonlocal resource, the superselection rule associated with particle number conservation leads to a new resource, the superselection induced variance of the local particle number. We show that, in the case of pure quantum states, one can quantify the nonlocal properties by only two additive measures, and that all states with the same measures can be asymptotically interconverted into each other by local operations and classical communication. Furthermore we discuss how superselection rules affect the concepts of majorization, teleportation, and mixed state entanglement.
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Affiliation(s)
- N Schuch
- Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, Garching, D-85748, Germany
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