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| Autori principali: | , , , , |
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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2402.18500 |
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| _version_ | 1866910196708671488 |
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| author | Alhambra, Álvaro M. Capel, Ángela Gondolf, Paul Ruiz-de-Alarcón, Alberto Scalet, Samuel O. |
| author_facet | Alhambra, Álvaro M. Capel, Ángela Gondolf, Paul Ruiz-de-Alarcón, Alberto Scalet, Samuel O. |
| contents | We show that spin chains in thermal equilibrium have a correlation structure in which individual regions are strongly correlated at most with their near vicinity. We quantify this with alternative notions of the conditional mutual information, defined through the so-called Belavkin-Staszewski relative entropy. We prove that these measures decay superexponentially at every positive temperature, under the assumption that the spin chain Hamiltonian is translation-invariant. Using a recovery map associated with these measures, we sequentially construct tensor network approximations in terms of marginals of small (sublogarithmic) size. As a main application, we show that classical representations of the states can be learned efficiently from local measurements with a polynomial sample complexity. We also prove an approximate factorization condition for the purity of the entire Gibbs state, which implies that it can be efficiently estimated to a small multiplicative error from a small number of local measurements. The results extend from strictly local to exponentially-decaying interactions above a threshold temperature, albeit only with exponential decay rates. As a technical step of independent interest, we show an upper bound to the decay of the Belavkin-Staszewski relative entropy upon the application of a conditional expectation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditional Independence of 1D Gibbs States with Applications to Efficient Learning Alhambra, Álvaro M. Capel, Ángela Gondolf, Paul Ruiz-de-Alarcón, Alberto Scalet, Samuel O. Quantum Physics Mathematical Physics We show that spin chains in thermal equilibrium have a correlation structure in which individual regions are strongly correlated at most with their near vicinity. We quantify this with alternative notions of the conditional mutual information, defined through the so-called Belavkin-Staszewski relative entropy. We prove that these measures decay superexponentially at every positive temperature, under the assumption that the spin chain Hamiltonian is translation-invariant. Using a recovery map associated with these measures, we sequentially construct tensor network approximations in terms of marginals of small (sublogarithmic) size. As a main application, we show that classical representations of the states can be learned efficiently from local measurements with a polynomial sample complexity. We also prove an approximate factorization condition for the purity of the entire Gibbs state, which implies that it can be efficiently estimated to a small multiplicative error from a small number of local measurements. The results extend from strictly local to exponentially-decaying interactions above a threshold temperature, albeit only with exponential decay rates. As a technical step of independent interest, we show an upper bound to the decay of the Belavkin-Staszewski relative entropy upon the application of a conditional expectation. |
| title | Conditional Independence of 1D Gibbs States with Applications to Efficient Learning |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2402.18500 |