Let $n\in\{1,2,\dots\}$ and let $C\in(0,+\infty)$. For each $i\in\{1,\dots,n\}$, let $(X_i,d_i)$ be a Polish [metric space](/page/Metric%20Space) with Borel $\sigma$-algebra $\mathcal B(X_i)$, and let $\rho_i\in\mathcal P(X_i)$. For probability measures $\alpha,\beta\in\mathcal P(X_i)$, define the extended quadratic Wasserstein distance by
if $\alpha\ll\rho_i$, and $\operatorname{Ent}_{\rho_i}(\alpha):=+\infty$ otherwise. Assume that each $\rho_i$ satisfies Talagrand's $T_2(C)$ inequality, namely
Equip $X$ with the product Borel $\sigma$-algebra, equivalently the Borel $\sigma$-algebra of the product Polish topology, and with the [product metric](/page/Product%20Metric) $d_2:X\times X\to[0,+\infty)$ defined by