Area laws and efficient descriptions of quantum many-body states
It is commonly believed that area laws for entanglement entropies imply that a quantum many-body state can be faithfully represented by efficient tensor network states—a conjecture frequently stated in the context of numerical simulations and analytical considerations. In this work, we show that thi...
| Hauptverfasser: | , |
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| Format: | Artikel |
| Sprache: | English |
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IOP Publishing
2016-01-01
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| Schriftenreihe: | New Journal of Physics |
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| Online Zugang: | https://doi.org/10.1088/1367-2630/18/8/083026 |
| _version_ | 1827873747196968960 |
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| author | Yimin Ge Jens Eisert |
| author_facet | Yimin Ge Jens Eisert |
| author_sort | Yimin Ge |
| collection | DOAJ |
| description | It is commonly believed that area laws for entanglement entropies imply that a quantum many-body state can be faithfully represented by efficient tensor network states—a conjecture frequently stated in the context of numerical simulations and analytical considerations. In this work, we show that this is in general not the case, except in one-dimension. We prove that the set of quantum many-body states that satisfy an area law for all Renyi entropies contains a subspace of exponential dimension. We then show that there are states satisfying area laws for all Renyi entropies but cannot be approximated by states with a classical description of small Kolmogorov complexity, including polynomial projected entangled pair states or states of multi-scale entanglement renormalisation. Not even a quantum computer with post-selection can efficiently prepare all quantum states fulfilling an area law, and we show that not all area law states can be eigenstates of local Hamiltonians. We also prove translationally and rotationally invariant instances of these results, and show a variation with decaying correlations using quantum error-correcting codes. |
| first_indexed | 2024-03-12T16:41:27Z |
| format | Article |
| id | doaj.art-66e6c38d6e9c495ca87f2ba4c6bbd0eb |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:41:27Z |
| publishDate | 2016-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-66e6c38d6e9c495ca87f2ba4c6bbd0eb2023-08-08T14:30:05ZengIOP PublishingNew Journal of Physics1367-26302016-01-0118808302610.1088/1367-2630/18/8/083026Area laws and efficient descriptions of quantum many-body statesYimin Ge0Jens Eisert1Max-Planck-Institut für Quantenoptik, D-85748 Garching, GermanyDahlem Center for Complex Quantum Systems, Freie Universität Berlin, D-14195 Berlin, GermanyIt is commonly believed that area laws for entanglement entropies imply that a quantum many-body state can be faithfully represented by efficient tensor network states—a conjecture frequently stated in the context of numerical simulations and analytical considerations. In this work, we show that this is in general not the case, except in one-dimension. We prove that the set of quantum many-body states that satisfy an area law for all Renyi entropies contains a subspace of exponential dimension. We then show that there are states satisfying area laws for all Renyi entropies but cannot be approximated by states with a classical description of small Kolmogorov complexity, including polynomial projected entangled pair states or states of multi-scale entanglement renormalisation. Not even a quantum computer with post-selection can efficiently prepare all quantum states fulfilling an area law, and we show that not all area law states can be eigenstates of local Hamiltonians. We also prove translationally and rotationally invariant instances of these results, and show a variation with decaying correlations using quantum error-correcting codes.https://doi.org/10.1088/1367-2630/18/8/083026entanglementarea lawstensor network statesKolmogorov complexity |
| spellingShingle | Yimin Ge Jens Eisert Area laws and efficient descriptions of quantum many-body states New Journal of Physics entanglement area laws tensor network states Kolmogorov complexity |
| title | Area laws and efficient descriptions of quantum many-body states |
| title_full | Area laws and efficient descriptions of quantum many-body states |
| title_fullStr | Area laws and efficient descriptions of quantum many-body states |
| title_full_unstemmed | Area laws and efficient descriptions of quantum many-body states |
| title_short | Area laws and efficient descriptions of quantum many-body states |
| title_sort | area laws and efficient descriptions of quantum many body states |
| topic | entanglement area laws tensor network states Kolmogorov complexity |
| url | https://doi.org/10.1088/1367-2630/18/8/083026 |
| work_keys_str_mv | AT yiminge arealawsandefficientdescriptionsofquantummanybodystates AT jenseisert arealawsandefficientdescriptionsofquantummanybodystates |