Let $(\mathcal A,\varphi)$ be a noncommutative probability space, where $\mathcal A$ is a unital algebra and $\varphi:\mathcal A\to\mathbb C$ is a unital linear functional. Let $(\mathcal A_i)_{i\in I}$ be a family of unital subalgebras of $\mathcal A$. For each $n\in\mathbb N$, let $\kappa_n:\mathcal A^n\to\mathbb C$ denote the $n$-th free cumulant determined by the moment-cumulant formula over noncrossing partitions. Then the family $(\mathcal A_i)_{i\in I}$ is free if and only if all mixed free cumulants vanish: for every integer $n\geq 2$, every choice of indices $i_1,\dots,i_n\in I$, and every choice of elements $a_j\in\mathcal A_{i_j}$ for $1\leq j\leq n$, \begin{align*} \{i_1,\dots,i_n\}\text{ has cardinality at least }2 \quad\Longrightarrow\quad \kappa_n(a_1,\dots,a_n)=0. \end{align*}