Towards Optimal Convergence Rates for the Quantum Central Limit Theorem

Abstract

The quantum central limit theorem for bosonic quantum systems states that the sequence of states $\ddot{I}{}^{⊞n}$ obtained from the $n$-fold convolution of a centered quantum state $\ddot{I}$ converges to a quantum Gaussian state $\ddot{I}{}_G$ that has the same first and second moments as $\ddot{I}$. 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 $\Vert \ddot{I}{}^{⊞n}-\ddot{I}{}_G{\Vert }_1=\mathcal{O}(n^{-1/2})$. We also introduce a notion of Poincaré inequality for quantum states and show that if $\ddot{I}$ satisfies this Poincaré inequality, then $D(\ddot{I}{}^{⊞n}\Vert \ddot{I}{}_G)=\mathcal{O}(n^{-1})$. By giving an explicit example, we verify that both these convergence rates are optimal.

43 pages, 1 figure. V2: Arguments have been improved. The proof of Lemma 13 has been revised. The term "Symmetric Lifting Map" has been introduced in place of "Tilde Maps," and the definition has been extended to include possibly unbounded arbitrary operators