Given two unitary matrices $A \in\mathcal{M}_{n\times n}=\{a_{ij}\}_{i,j \in {1..n}}$ and $B \in\mathcal{M}_{p\times p}=\{b_{ij}\}_{i,j \in {1..p}}$ where n>p.
Can I say that $\min(|\{a_{ij}\}_{i,j \in {1..n}}|)\leq \min(|\{b_{ij}\}_{i,j \in {1..p}}|)$?
Counter-example using real matrices : Let $A=Id\in \mathcal{M}_{3\times3}$. Then $A$ is trivially unitary. Now let $$B=\begin{pmatrix} cos(\theta) && -sin(\theta)\\ sin(\theta) && cos(\theta) \end{pmatrix}$$
Be the rotation matrix (thus orthogonal, which is a special case of unitary) of angle $\theta$ in the plane. For $\theta\in (0,\frac{\pi}{2})$, both $|cos(\theta)|$ and $|sin(\theta)|$ are smaller than one, so your inequality wont hold.