Consider matrix subspace $\mathbb{C}^{m\times n}$. What is the minimum $d$ such that for any $d$-dimensional matrix subspace of $\mathbb{C}^{m\times n}$, there exists two distinct matrix $M,N$ such that $$ M^{+}M=N^{+}N $$ here $M^{+}$ denotes the hermitian conjugate of $M$.

The distinct means that there is no $\lambda$ such that $M=\lambda N$.