Towards Optimal Convergence Rates for the Quantum Central Limit Theorem

• URL: http://arxiv.org/abs/2310.09812v1
• Date: Sun, 15 Oct 2023 12:02:43 GMT
• Title: Towards Optimal Convergence Rates for the Quantum Central Limit Theorem
• Authors: Salman Beigi, Hami Mehrabi
• Abstract summary: Quantum central limit theorem for bosonic systems states that the sequence of states $rhoboxplus n$ obtained from the $n$-fold convolution of a centered quantum state converges to a quantum Gaussian state. In this paper, we contribute to the problem of finding optimal rate of convergence for this theorem.
• Score: 3.6985338895569204
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Quantum central limit theorem for bosonic systems states that the sequence of states $\rho^{\boxplus n}$ obtained from the $n$-fold convolution of a centered quantum state $\rho$ converges to a quantum Gaussian state $\rho_G$ that has the same first and second moments as $\rho$. In this paper, we contribute to the problem of finding optimal rate of convergence for this theorem. We first show that if an $m$-mode quantum state has a finite moment of order $\max\{3, 2m\}$, then we have $\|\rho - \rho_G\|_1=\mathcal O(n^{-1/2})$. By giving an explicit example, we verify that this convergence rate is optimal. We also introduce a notion of Poincar\'e inequality for quantum states and show that if $\rho$ satisfies this Poincar\'e inequality, then $D(\rho\| \rho_G)= \mathcal O(n^{-1})$.

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