Entropy on the Configuration Space: Tensorization, Subadditivity over the Particles, and the One-Particle Marginal

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used (the letter $P$ below denotes a probability measure on a configuration space), let $q,N\in\mathbb{N}$. Block maps $\mathfrak{p}_{k}$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks, tensor powers $\rho^{\otimes N}$ those of The Tensor Power of a Probability Measure on Euclidean Space §tensor, and one-particle marginals $P^{[1]}$ those of The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal; $\mathrm{Ent}$ and $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{m})$ are as in The Entropy of a Probability Measure on Euclidean Space §entropy, in dimensions $m=q$ and $m=qN$.

1. (Tensor powers)¶ For $\rho\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$, $\rho^{\otimes N}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{qN})$ and $\mathrm{Ent}(\rho^{\otimes N})=N\,\mathrm{Ent}(\rho)$.

2. (Subadditivity)¶ For $P\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{qN})$, each particle law $(\mathfrak{p}_{k})_{\#}P$ belongs to $\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$, and $\sum_{k=1}^{N}\mathrm{Ent}\bigl((\mathfrak{p}_{k})_{\#}P\bigr)\le\mathrm{Ent}(P)$.

3. (One-particle marginal)¶ For $P\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{qN})$, $P^{[1]}\in\mathcal{P}_{2}^{\mathrm{Ent}}(\mathbb{R}^{q})$ and $N\,\mathrm{Ent}(P^{[1]})\le\mathrm{Ent}(P)$.