Is treatment conditionally independent from outcome in Single Experiment Design? I'm reading this slides.
At slide 10 there is written that in "Single Experiment Design" we assume "Randomization of treatment", that is:
$ \{ Y_i(t,m),M_i(t') \} \perp  T_i \lvert X $
I don't understand how the outcome and the treatment can be conditionally independent given X, as long as the treatment has an effect on the outcome.
Here there is written that:

The rule here is that after you've drawn out the graph, two events are conditionally independent if you can't traverse from one node to the other without going through a "blocked" node, where a "blocked" node is an event that has has already happened.

But it looks like, if the treatment has an effect on the outcome, I can go from treatment to outcome without going through the X.
What am I not understanding?
I think he wants to say that the treatment is random, meaning there are no unobserved covariates that determine both treatment and outcome.
But this seems not to imply that treatment and outcome are conditionally independent of each other given X.
Or maybe he is abusing notation?

This is what I think I was missing:

If the treatment assignment is randomized, as in the Brader, Valentino, and Suhay (2008) study, then the treatment is jointly independent of the potential outcomes because the probability of receiving the treatment is identical regardless of the values of the potential outcomes. We can write this as {Yi(1), Yi(0)}⊥⊥ Ti with the standard symbol of statistical independence.

So it's not treatment that is not correlated with outcome, it's the probability of receiving treatment that does not depend on the value of potential outcome.
Reference:
Imai, K., Keele, L., Tingley, D., Yamamoto, T., 2011. Unpacking the Black Box of Causality: Learning about Causal Mechanisms from Experimental and Observational Studies. Am Polit Sci Rev 105, 765–789. https://doi.org/10.1017/S0003055411000414
