They are very much distinguishable and no longer one person. They are the you that willed after the split but now 2 physically different people.
We only know so far that it is objectively random; we don't know that there is not a "subjective force" piloting QM.
We don't KNOW that there's not a teapot orbiting the sun between Jupiter and Mars either. But we are best served by assuming that there is not.
Nobody is claiming this is scientific (although Hameroff and Penrose do
https://www.youtube.com/watch?v=ztGNznlowic ). If you are going to stay in the scientific realism domain, then you can't even say it is false.
I am responding to the seemingly negatively certain OP and saying that there may be ways we can still have free will at least in general philosophically/logically.
Hang on there! I'm not saying that we don't have free will. I'm saying that there isn't a freewill problem. More specifically, I'm asserting that perhaps even thinking about freewill as freewill isn't terribly helpful.
Indeed, many of the shibboleths of freewill just seem to be accepted uncritically and are rather difficult to think around.
We are not and never could be predictable by any Laplacian demon: the world is stochastic, our brain is stochastic and there exist clear mathematical proofs of irreducible emergent complexity. Not least the Banach Tarski paradox and, of course, the good old sensitive dependence on initial conditions of Chaos theory. You just need one single example of a genuine indeterminacy, as provided by Bell's inequality, and Laplace's demon is blind. We can do
slightly better than that. Obviously, the demon is just a metaphor for the world itself, but the world is not and never has been purely deterministic.
'Could have done otherwise all other conditions remaining the same', is still under debate here, but I'm still not quite sure why anyone who isn't after a deity (although that was a good point) would take the idea terribly seriously.
'freedom from causal constraint', Always reminds me of Pinball Wizard from Tommy:
https://www.youtube.com/watch?v=EK33CY68s1w
Freedom from causation is freedom from perception. That's a high cost for freewill and would leave you unable to respond to anything.
Finally, the point at which we locate freewill appears oddly difficult to pin down. Descartes' captain is out and that leaves the field rather open...
maintaining an obsession with Laplacian notions of predictability and so on while ignoring simple biological facts like the reality and inherent necessity of stochastic neural function.
We don't