Let $U_1,...,U_n$ and $V_1,...,V_n$ be two sets of $n$ unitary matrices of the same size. We'll denote $E(U_i,V_i)= \max_v \, |(U_i -V_i)v|$ (max over all the quantum states), $U=\prod_i U_i$ and $V=\prod_i V_i$.
I'd like to show that $E(U,V) \leq \sum_i E(U_i,V_i) $.
At first I thought proving this by induction on $n$, but then I got stuck even in the simple case of $n=2$.
I also tried expanding the expression: $$E(U,V)=\max_v |(\prod U_i - \prod V_i)v | $$
But I got stuck on here too. Maybe there's an easier way I'm missing out?
Edit: $U_i, V_i$ are unitary matrices
Let $||.||$ be the matrix norm induced by the $l_2$ norm over $\mathbb{C}^n$. Then (if I correctly understand the question) $E(U,V)=||U-V||$. Note that, if $U$ is unitary, then $||U||=1$ and that the set of unitary matrices is a group.
Case $n=2$. $||U_1U_2-V_1V_2||=||(U_1-V_1)U_2+V_1(U_2-V_2)||\leq ||U_1-V_1||||U_2||+||V_1||||U_2-V_2||$ and we are done.
Case $n=3$. According to the previous calculation, $||U_1U_2U_3-V_1V_2V_3||\leq ||U_1U_2-V_1V_2||+||U_3-V_3||\leq ||U_1-V_1||+||U_2-V_2||+||U_3-V_3||$.
Case $n>3$. And so on...