For 4 independent random variables $A, B, C, D$ and Wasserstein-1 distance $W^1$,
$W^1(P_{A,B} \parallel P_{C,D})=W^1(P_A \parallel P_C)+W^1(P_B \parallel P_D)$
Does the above equation generally hold?
$P_{A,B}$ is the independent coupling of $A$ and $B$, i.e. $P_{A,B}=P_A \cdot P_B$ and $P_{C,D}$ is similarly defined.
My observation is that $$W^1(P_{A,B} \parallel P_{C,D})=\inf_{\pi \in \Pi(P_{A,B},P_{C,D})}E_{a,b,c,d\sim{\pi}}\|a-c\|_1+E_{a,b,c,d\sim{\pi}}\|b-d\|_1$$
I want to know if this is equal to: $$=\inf_{\phi \in \Pi(P_A, P_C)}E_{a,c \sim{\phi}}\|a-c\|_1+\inf_{\phi \in \Pi(P_B, P_D)}E_{b,d \sim{\phi}}\|b-d\|_1$$
where $\Pi(\cdot, \cdot)$ denotes all possible couplings between two distributions.
Thank you