Let $n\in\mathbb N$, let $p\in[1,\infty)$, and for each $i\in\{1,\dots,n\}$ let $(X_i,d_i)$ be a Polish metric space equipped with its Borel $\sigma$-algebra $\mathcal B(X_i)$. Let $\mathcal P(X_i)$ denote the set of Borel probability measures on $X_i$, let $\Pi(\mu_i,\eta_i)$ denote the set of Borel probability couplings of $\mu_i,\eta_i\in\mathcal P(X_i)$, and let $\rho_i\in\mathcal P(X_i)$. Assume that, for each $i\in\{1,\dots,n\}$, there exists a constant $C_i\in(0,\infty)$ such that every $\nu_i\in\mathcal P(X_i)$ satisfies

\begin{align*} W_{d_i,p}(\nu_i,\rho_i)^p\le C_i H(\nu_i\mid\rho_i). \end{align*}

Here

\begin{align*} W_{d_i,p}(\nu_i,\rho_i)^p:=\inf_{\pi\in\Pi(\nu_i,\rho_i)}\int_{X_i\times X_i} d_i(x_i,y_i)^p\,d\pi((x_i,y_i)), \end{align*}

and $H(\nu_i\mid\rho_i)$ denotes relative entropy, with value $+\infty$ when $\nu_i\not\ll\rho_i$.

Set $X:=\prod_{i=1}^n X_i$, equip $X$ with its product Borel $\sigma$-algebra, let $\mathcal P(X)$ denote the set of Borel probability measures on $X$, and set $\rho:=\bigotimes_{i=1}^n \rho_i$. Define $d_p:X\times X\to[0,\infty)$ by

\begin{align*} d_p(x,y):=\left(\sum_{i=1}^n d_i(x_i,y_i)^p\right)^{1/p} \end{align*}

for $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ in $X$. For $\mu,\eta\in\mathcal P(X)$, let $\Pi(\mu,\eta)$ denote the set of Borel probability couplings of $\mu$ and $\eta$. Then every $\nu\in\mathcal P(X)$ satisfies

\begin{align*} W_{d_p,p}(\nu,\rho)^p\le\left(\max_{1\le i\le n} C_i\right)H(\nu\mid\rho), \end{align*}

where

\begin{align*} W_{d_p,p}(\nu,\rho)^p:=\inf_{\Gamma\in\Pi(\nu,\rho)}\int_{X\times X} d_p(x,y)^p\,d\Gamma((x,y)). \end{align*}