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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.00743 |
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| _version_ | 1866918128643997696 |
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| author | Toniolo, Daniele Bose, Sougato |
| author_facet | Toniolo, Daniele Bose, Sougato |
| contents | We consider a generic one dimensional spin system of length $ L $, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical $ α$-Rényi entropies, $ 0 < α\le 1 $, of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical $ α$-Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of $α$ close enough to $ 1 $, and moreover to initial pure states with low entanglement, of order $ \log L $, that are typically represented by critical states. We establish that low entanglement states have an efficient MPS representation that persists at least up to times of order $ \log L $. The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on $ α$-Rényi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00743 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The dynamical $α$-Rényi entropies of local Hamiltonians grow at most linearly in time Toniolo, Daniele Bose, Sougato Quantum Physics Mathematical Physics We consider a generic one dimensional spin system of length $ L $, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical $ α$-Rényi entropies, $ 0 < α\le 1 $, of an initial product state grow at most linearly in time. This result arises from a general relation among dynamical $ α$-Rényi entropies and Lieb-Robinson bounds. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of $α$ close enough to $ 1 $, and moreover to initial pure states with low entanglement, of order $ \log L $, that are typically represented by critical states. We establish that low entanglement states have an efficient MPS representation that persists at least up to times of order $ \log L $. The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on $ α$-Rényi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest. |
| title | The dynamical $α$-Rényi entropies of local Hamiltonians grow at most linearly in time |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2408.00743 |