Consider the $W^2$ Wasserstein distance on $\Bbb{R}^2$, which we take with its Euclidean metric.
Given a probability measure $p$ on $\Bbb{R}$, consider the following two couplings of $p$ with itself:
- The identity coupling $\mathrm{id}_p$ on $\Bbb{R}^2$, supported entirely on the diagonal {$(x,x):x\in\Bbb{R}$};
- The product coupling $p\otimes p$ on $\Bbb{R}^2$.
Is there an easy way to calculate, as measures on $\Bbb{R}^2$, the $W^2$ distance between $\mathrm{id}_p$ and $p\otimes p$?