A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory
Abstract Unitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approxima...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer International Publishing
2022
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| Online Access: | https://hdl.handle.net/1721.1/144160 |
| _version_ | 1826199024594583552 |
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| author | Ranard, Daniel Walter, Michael Witteveen, Freek |
| author2 | Massachusetts Institute of Technology. Center for Theoretical Physics |
| author_facet | Massachusetts Institute of Technology. Center for Theoretical Physics Ranard, Daniel Walter, Michael Witteveen, Freek |
| author_sort | Ranard, Daniel |
| collection | MIT |
| description | Abstract
Unitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb–Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb–Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb–Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest. |
| first_indexed | 2024-09-23T11:13:32Z |
| format | Article |
| id | mit-1721.1/144160 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:13:32Z |
| publishDate | 2022 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1441602023-12-21T22:14:16Z A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory Ranard, Daniel Walter, Michael Witteveen, Freek Massachusetts Institute of Technology. Center for Theoretical Physics Abstract Unitary dynamics with a strict causal cone (or “light cone”) have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb–Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse to the Lieb–Robinson bounds: any ALPU of index zero can be exactly generated by some time-dependent, quasi-local Hamiltonian in constant time. For the special case of finite chains with open boundaries, any unitary satisfying the Lieb–Robinson bound may be generated by such a Hamiltonian. We also discuss some results on the stability of operator algebras which may be of independent interest. 2022-08-01T12:15:58Z 2022-08-01T12:15:58Z 2022-07-26 2022-07-31T03:12:00Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/144160 Ranard, Daniel, Walter, Michael and Witteveen, Freek. 2022. "A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory." PUBLISHER_CC en https://doi.org/10.1007/s00023-022-01193-x Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Ranard, Daniel Walter, Michael Witteveen, Freek A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory |
| title | A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory |
| title_full | A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory |
| title_fullStr | A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory |
| title_full_unstemmed | A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory |
| title_short | A Converse to Lieb–Robinson Bounds in One Dimension Using Index Theory |
| title_sort | converse to lieb robinson bounds in one dimension using index theory |
| url | https://hdl.handle.net/1721.1/144160 |
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