Let $\mathbb{C}\langle X_1,\dots,X_d\rangle$ be the unital complex noncommutative polynomial $*$-algebra with involution determined by $X_i^* = X_i$ for each $i \in \{1,\dots,d\}$, conjugation on coefficients, and reversal of words. Let $\mu: \mathbb{C}\langle X_1,\dots,X_d\rangle \to \mathbb{C}$ be a complex-linear functional satisfying $\mu(1)=1$.

Then $\mu$ is the joint law of $d$ self-adjoint elements in an algebraic noncommutative probability space if and only if, for every $P \in \mathbb{C}\langle X_1,\dots,X_d\rangle$,

\begin{align*} \mu(P^*P) \in [0,\infty). \end{align*}