I'm new to causal inference, and I have a question regarding a potential causal relationship. I'm wondering if the following assumption holds:
$(Y_i(1), Y_i(0), X_i) ⊥ Z_i -> (Y_i(1), Y_i(0)) ⊥ Z_i | X_i$
I want to know if the independence between the treatment assignment ($Y_i$(1), $Y_i$(0)), the outcome variable ($Z_i$), and the covariate ($X_i$) implies that the conditional independence of (Yi(1), Yi(0)) and $Z_i$ holds, given $X_i$.
No, it is not necessarily the case that marginal independence implies conditional independence generally.
As a counter-example consider the following path in a causal diagram $$ Z \leftarrow U_1 \rightarrow X \leftarrow U_2 \rightarrow Y$$ Here, $Z$ and $Y(z)$ are marginally independent, but conditioning on $X$ opens this path, so $Z$ and $Y(z)$ are not independent conditional on $X$.
In the special case of marginally randomized experiments, where $Z$ is assigned randomly without reference to any covariates, then marginal independence will imply conditional independence for any covariates $X$ that occur prior in time to $Z$, as you have written above. This is because there is no arrows pointing into $Z$, so conditioning on any $X$ that occurs prior to $Z$ can't open a backdoor path between $Z$ and $Y$. If $X$ occurs after $Z$, the independence statement is also not necessarily true (e.g., $X$ could be on a causal path between $Z$ and $Y$, causing other issues).
