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 \(\rho _\textrm{G}\) that has the same first and second moments as \(\rho \) . In this paper, we contribute to resolving 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 \(\Vert \rho ^{\boxplus n} - \rho _\textrm{G}\Vert _1={\mathcal {O}}(n^{-1/2})\) . We also introduce a notion of Poincaré inequality for quantum states and show that if \(\rho \) satisfies this Poincaré inequality, then \(D(\rho ^{\boxplus n}\Vert \rho _\textrm{G})= {\mathcal {O}}(n^{-1})\) . By giving an explicit example, we verify that both these convergence rates are optimal.
