If we have the following upper bound on the sum of trace distances:
$$ \frac{1}{N} \sum_{a, b}||p_1(a | b) \rho_{ab} - p_2(a | b) \sigma_{ab}|| \le \epsilon, $$ where $p_1$ and $p_2$ are two probability distributions and $\rho_{ab}$ and $\sigma_{ab}$ are quantum states depending on $a, b$. Then, what is the best that we could say about the following in terms of $\epsilon$:
$$ ||\rho_{ab} - \sigma_{ab}|| \le ? $$