Let $d\in \mathbb{N}$ and $\gamma$ be a probability measure on $\mathbb{R}^{2d}$ (say with finite second moments).
Assume that $\gamma$ is concentrated on the diagonal $(x=y), x,y \in \mathbb{R}^d$, i.e $\gamma(A\times B)=\gamma(A\cap B \times A\cap B)~~$for all measureable $A,B \subset \mathbb{R}^d$.
What can we say about the entropy $H$ of $\gamma$ ?
$\textbf{Edit : (some context)}$
The reason I am asking is that I want an upper bound for the (entropy regularised) Wasserstein operator between a distribution and itself i.e define the entropy regularised Wasserstein (between densities $\mu$ and $\nu$ as $$ \tilde{W}_{2}^2(\mu,\nu):=\inf_{\gamma \in \Pi(\mu,\nu)} \int_{\mathbb{R}^{2d}}|x-y|^2 \gamma(dx,dy)+\int_{\mathbb{R}^{2d}} \gamma \log \gamma. $$ Where $\Pi$ is the space of joint densities with marginals $\mu,\nu$. Then I was looking for an upper bound on $\tilde{W}^2_2(\mu,\mu)$.