Entanglement, quantum randomness, and complexity beyond scrambling
Abstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quant...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-07-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | http://link.springer.com/article/10.1007/JHEP07(2018)041 |
| _version_ | 1819103426018017280 |
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| author | Zi-Wen Liu Seth Lloyd Elton Zhu Huangjun Zhu |
| author_facet | Zi-Wen Liu Seth Lloyd Elton Zhu Huangjun Zhu |
| author_sort | Zi-Wen Liu |
| collection | DOAJ |
| description | Abstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom. This paper is an extended version of Phys. Rev. Lett. 120 (2018) 130502 [1]. |
| first_indexed | 2024-12-22T01:50:16Z |
| format | Article |
| id | doaj.art-a89ca9f078ed4cffbe7a124d246ca531 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-22T01:50:16Z |
| publishDate | 2018-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-a89ca9f078ed4cffbe7a124d246ca5312022-12-21T18:42:57ZengSpringerOpenJournal of High Energy Physics1029-84792018-07-012018716210.1007/JHEP07(2018)041Entanglement, quantum randomness, and complexity beyond scramblingZi-Wen Liu0Seth Lloyd1Elton Zhu2Huangjun Zhu3Center for Theoretical Physics, Massachusetts Institute of TechnologyDepartment of Physics, Massachusetts Institute of TechnologyCenter for Theoretical Physics, Massachusetts Institute of TechnologyInstitute for Theoretical Physics, University of CologneAbstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom. This paper is an extended version of Phys. Rev. Lett. 120 (2018) 130502 [1].http://link.springer.com/article/10.1007/JHEP07(2018)041Random SystemsBlack Holes in String TheoryHolography and condensed matter physics (AdS/CMT)Stochastic Processes |
| spellingShingle | Zi-Wen Liu Seth Lloyd Elton Zhu Huangjun Zhu Entanglement, quantum randomness, and complexity beyond scrambling Journal of High Energy Physics Random Systems Black Holes in String Theory Holography and condensed matter physics (AdS/CMT) Stochastic Processes |
| title | Entanglement, quantum randomness, and complexity beyond scrambling |
| title_full | Entanglement, quantum randomness, and complexity beyond scrambling |
| title_fullStr | Entanglement, quantum randomness, and complexity beyond scrambling |
| title_full_unstemmed | Entanglement, quantum randomness, and complexity beyond scrambling |
| title_short | Entanglement, quantum randomness, and complexity beyond scrambling |
| title_sort | entanglement quantum randomness and complexity beyond scrambling |
| topic | Random Systems Black Holes in String Theory Holography and condensed matter physics (AdS/CMT) Stochastic Processes |
| url | http://link.springer.com/article/10.1007/JHEP07(2018)041 |
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