  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 $D(\rho\| \rho_G)= \mathcal
O(n^{-1})$.
