Let us define a distance metric between two joint probability math functions $p(x,y)$ and $q(x,y)$ as in the following

\begin{align} \sum_{y}\sqrt{\sum_{x}p(x)\left(p(y|x)-q(y|x)\right)^2}. \end{align}

By Jensen inequality, we have

\begin{align*} \sum_{y}\sqrt{\sum_{x}p(x)\left(p(y|x)-q(y|x)\right)^2}&\geq\sum_{x,y}\lvert p(x,y)-q(x,y)\rvert\\ &=\lVert p(x,y)-q(x,y)\rVert_1 \end{align*}

where $\lVert\cdot\rVert_1$ is the total variation distance.

What is the name of this distance? Is there any other relationship between this distance and any other stochastic distances?