Towards Optimal Convergence Rates for the Quantum Central Limit Theorem

The quantum central limit theorem for bosonic quantum 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 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 $\|\rho^{\boxplus n} - \rho_G\|_1=\mathcal O(n^{-1/2})$. We also introduce a notion of Poincar\'e inequality for quantum states and show that if $\rho$ satisfies this Poincar\'e inequality, then $D(\rho^{\boxplus n}\| \rho_G)= \mathcal O(n^{-1})$. By giving an explicit example, we verify that both these convergence rates are optimal.