Optimal convergence rates in trace distance and relative entropy for the quantum central limit theorem

Abstract: A quantum analogue of the Central Limit Theorem (CLT), first introduced by Cushen and Hudson (1971), states that the $n$-fold convolution $\rho^{\boxplus n}$ of an $m$-mode quantum state $\rho$ with zero first moments and finite second moments converges weakly, as $n$ increases, to a Gaussian state $\rho_G$ with the same first and second moments. Recently, this result has been extended with estimates of the convergence rate in various distance measures. In this paper, we establish optimal rates of convergence in both the trace distance and quantum relative entropy. Specifically, we show that for a centered $m$-mode quantum state with finite third-order moments, the trace distance between $\rho^{\boxplus n}$ and $\rho_G$ decays at the optimal rate of $\mathcal{O}(n^{-1/2})$, consistent with known convergence rates. Furthermore, for states with finite fourth-order moments (plus a small correction $\delta$ for $m>1$), we prove that the relative entropy between $\rho^{\boxplus n}$ and $\rho_G$ decays at the optimal rate of $\mathcal{O}(n^{-1})$. Both of these rates are proven to be optimal, even when assuming the finiteness of all moments of $\rho$. These results relax previous assumptions on higher-order moments, yielding convergence rates that match the best known results in the classical setting. Our proofs draw on techniques from the classical literature, including Edgeworth-type expansions of quantum characteristic functions, adapted to the quantum context. A key technical step in the proof of our entropic CLT is establishing an upper bound on the relative entropy distance between a general quantum state and its Gaussification. As a by-product of this, an upper bound on the relative entropy of non-Gaussianity is derived, which is of independent interest.