Let $(\mathcal A,\varphi)$ be a positive tracial noncommutative probability space, where $\mathcal A$ is a unital $*$-algebra and $\varphi:\mathcal A\to\mathbb C$ is a positive unital linear functional satisfying $\varphi(ab)=\varphi(ba)$ for all $a,b\in\mathcal A$. Let $(s_1,\dots,s_d)$ be a self-adjoint semicircular family in $\mathcal A$, and let $C=(C_{ij})_{1\leq i,j\leq d}$ be its covariance matrix, defined by

\begin{align*} C_{ij}=\varphi(s_i s_j) \end{align*}

for all $1\leq i,j\leq d$. Then $C$ is a real symmetric positive semidefinite matrix; equivalently, for every $a=(a_1,\dots,a_d)\in\mathbb R^d$,

\begin{align*} \sum_{i=1}^{d}\sum_{j=1}^{d} a_i C_{ij} a_j \geq 0. \end{align*}