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 ${\rho }^{⊞n}$ 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 $\mathrm{\parallel }{\rho }^{⊞n}-{\rho }_G{\mathrm{\parallel }}_1=\mathcal{O}(n^{-1\mathrm{/}2})$. We also introduce a notion of Poincaré inequality for quantum states and show that if $\rho$ satisfies this Poincaré inequality, then $D({\rho }^{⊞n}\mathrm{\parallel }{\rho }_G)=\mathcal{O}(n^{-1})$. By giving an explicit example, we verify that both these convergence rates are optimal.