Total Variation Distance between Two Matrices

Question

I have two $n \times n$ matrices $P$ and $Q$. They are given as follows:

$$P = \begin{bmatrix}p_{11}&p_{12}&........&p_{1n}\\p_{21}&p_{22}&........&p_{2n}\\...&...&........&...\\...&...&........&...\\p_{n1}&p_{n2}&........&p_{nn}\end{bmatrix}$$

$$Q = \begin{bmatrix}q_{11}&q_{12}&........&q_{1n}\\q_{21}&q_{22}&........&q_{2n}\\...&...&........&...\\...&...&........&...\\q_{n1}&q_{n2}&........&q_{nn}\end{bmatrix}$$

Now, I know that the Total Variation Distance between $P$ and $Q$ is at most $\delta$, i.e., $d_{TV}(P,Q) \leq \delta$. Is there any way I can relate this $\delta$ to the difference between each entry, for example $(p_{11} - q_{11})$, or $(p_{12} - q_{12})$, and so on?

Answer 1

No. In general, the best you can say is that $|p_{ij} - q_{ij}| \leq 2\delta$ for all $i,j$. This is clearly tight: take $P = 2\delta e_i e_j^T$ and $Q = 0$.

(Note: this is a consequence of the general fact that a $\ell_1$ norm implies a very pessimistic bound on the $\ell_\infty$ norm.)