Let $P$ and $Q$ be two probability measures on $\{1,\dots,n\}$. Are there references regarding $$d_{TV}(P\otimes Q,Q\otimes P)\,,$$ where $d_{TV}$ is the total variation distance?
In other words, if we represent $P$ and $Q$ by two $n$-dimensional vectors $p,q\in \mathbb{R}^n$, the above expression is $$ \frac{1}{2}\|pq^\top-qp^\top\|_1\,, $$ where $\|A\|_1$ is the $\ell^1$ norm of $\operatorname{vec}(A)$ (i.e., the flattened version of $A$).