In linear algebra, it is well-known that the rank of a matrix is subadditive, that is, for any matrices A and B, we have rank(A + B) ≤ rank(A) + rank(B). This property leads to a natural question: for any polynomial p over non-commuting indeterminates x and y, is there any natural upper bound for rank(p(A,B)) with matrices A and B not totally deterministic a prior? Furthermore, can this upper bound be achieved in a dimensionless optimal sense?

We address these questions with the aid of free probability theory, which will be briefly introduced in this talk. Our findings are grounded in the universal property of freely independent random variables with respect to the von Neumann rank. This universality reveals that the free independent copy of matrices can provide an upper bound on the rank. Interestingly, this upper bound can be saturated by matrices, owing to the relationship between free probability and random matrices. This talk is based on a joint-work with Octavio Arizmendi, Guillaume C´ebron and Roland Speicher.