$\DeclareMathOperator\sgn{sgn}$Suppose A is a $N \times N$ Hermitian and unitary matrix, i.e., $A^{\dagger}=A$ and $A^{\dagger}A=I =AA^{\dagger}$. (Assume all entries are real.)

And let $u \in \{-1,1\}^N$, $v \in \{-1,1\}^N$.

Suppose $\|u- v\|_1 \leq \epsilon N$ (i.e., $u$ and $v$ differ on $\epsilon N$ coordinates) .

and $\sgn: \mathbb{R} \rightarrow \{-1,0,1\}$ be the sign function, i.e., maps all negative numbers to $-1$, and positive numbers to $1$, and $0$ to $0$.

I want a non-trivial upper bound on $$\|\sgn(Au)-\sgn(Av)\|_1$$ in terms of $\epsilon$. For example, is it upper bounded by $4\epsilon N$?