Let $(\mathcal A,\varphi)$ be a noncommutative probability space, and let $(\mathcal A_i)_{i \in I}$ be a family of unital subalgebras of $\mathcal A$. For each $n \in \mathbb N$, let

\begin{align*} \kappa_n: \mathcal A^n \to \mathbb C \end{align*}

denote the $n$-th free cumulant functional determined by the moment-cumulant formula over noncrossing partitions. Then the family $(\mathcal A_i)_{i \in I}$ is freely independent if and only if the following mixed-cumulant condition holds: for every $n \geq 2$, every choice of indices $i_1,\dots,i_n \in I$ not all equal, and every choice of elements $a_j \in \mathcal A_{i_j}$ for $1 \leq j \leq n$, one has

\begin{align*} \kappa_n(a_1,\dots,a_n)=0. \end{align*}