Towards Optimal Convergence Rates for the Quantum Central Limit Theorem
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912512158466048 |
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| author | Beigi, Salman Mehrabi, Hami |
| author_facet | Beigi, Salman Mehrabi, Hami |
| contents | The quantum central limit theorem for bosonic quantum systems states that the sequence of states $ρ^{\boxplus n}$ obtained from the $n$-fold convolution of a centered quantum state $ρ$ converges to a quantum Gaussian state $ρ_G$ that has the same first and second moments as $ρ$. In this paper, we contribute to the problem of finding the optimal rate of convergence for this quantum central limit theorem. We first show that if an $m$-mode quantum state has a finite moment of order $\max\{3, 2m\}$, then we have $\|ρ^{\boxplus n} - ρ_G\|_1=\mathcal O(n^{-1/2})$. We also introduce a notion of Poincaré inequality for quantum states and show that if $ρ$ satisfies this Poincaré inequality, then $D(ρ^{\boxplus n}\| ρ_G)= \mathcal O(n^{-1})$. By giving an explicit example, we verify that both these convergence rates are optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09812 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards Optimal Convergence Rates for the Quantum Central Limit Theorem Beigi, Salman Mehrabi, Hami Quantum Physics The quantum central limit theorem for bosonic quantum systems states that the sequence of states $ρ^{\boxplus n}$ obtained from the $n$-fold convolution of a centered quantum state $ρ$ converges to a quantum Gaussian state $ρ_G$ that has the same first and second moments as $ρ$. In this paper, we contribute to the problem of finding the optimal rate of convergence for this quantum central limit theorem. We first show that if an $m$-mode quantum state has a finite moment of order $\max\{3, 2m\}$, then we have $\|ρ^{\boxplus n} - ρ_G\|_1=\mathcal O(n^{-1/2})$. We also introduce a notion of Poincaré inequality for quantum states and show that if $ρ$ satisfies this Poincaré inequality, then $D(ρ^{\boxplus n}\| ρ_G)= \mathcal O(n^{-1})$. By giving an explicit example, we verify that both these convergence rates are optimal. |
| title | Towards Optimal Convergence Rates for the Quantum Central Limit Theorem |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2310.09812 |