Quantum central limit theorem for self-adjoint non-commutative polynomials

Let $(\mathcal{M},\rho)$ be a quantum probability space, and let $A_1,\ldots,A_k$ be self-adjoint elements with mean $0$ and symmetric covariance matrix. For each $t$, write

$$A_a^{(t)}=I^{\otimes t-1}\otimes A_a\otimes I^{\otimes N-t-1},$$

and define

$$\widetilde{A}_a=\frac{1}{\sqrt{N}}\sum_{t=1}^N A_a^{(t)}.$$

Let $p\in\mathbb{C}\langle A_1,\ldots,A_k\rangle$ be a self-adjoint non-commutative polynomial, and let $\widehat{p}$ be its commutative image. Quantum central limit conjecture. The distribution of $p(\widetilde{A}_1,\ldots,\widetilde{A}_k)$ converges as $N\to\infty$ to the distribution of $\widehat{p}(X_1,\ldots,X_k)$, where $X_1,\ldots,X_k$ are classical Gaussian random variables with covariance matrix

$$E[X_aX_b]=\rho(A_aA_b).$$

In the tracial case, the symmetric covariance makes the limiting law classical and suggests that the non-commutative polynomial contributes only through its commutative image; the conjecture addresses the non-commutative obstruction in the multivariate quantum central limit theorem.