The linked article by Elizer Yudkowsky is straight up wrong for the following reasons:

(1) Eliezer's understanding of the physics here is bunk. I'm actually a trained physist. He is not. But bonus points to you if you reject this argument because you shouldn't accept my authority any more than you should accept his. I assume you read Griffiths' Quantum Mechanics or a similar introductory book and came to your own conclusions?

(2) Specifically the experimental result Eliezer quotes has to do with how we calculate probabilities for quantum mechanical events. There are an infinitely many ways one could calculate probabilities -- math describes the universe, it doesn't constrain it. But if you do so naively, you end up with one answer if you treat "P1 at L1, P2 at L2" as a different state than "P1 at L2, P2 at L1" than if you treat them as the same state. Experimental results show that the latter probabilities are correct. One interpretation is that P1 and P2 are the same particle, so the state is "P at L1, P at L2". That's one interpretation. Another perfectly valid interpretation is that "Particle of type

at L1, Particle of type

at L2" is the actual state -- that is to say that the particles keep their identity but identity doesn't factor into the probabalistic calculus. That's why the term used by phsyisits is distinguishable rather than identity. These particles are indistinguishable, but that does not mean they are identical. That would be an unwaranted inference.

(3) All of that is a moot point, because it doesn't match up at all with what we are talking about: the continuity of self as it relates to human minds. Calculating probabilities about particles in boxes tells us nothing about whether I would expect to wake up in a computer after a destructive upload, or how that relates to a personal desire to cheat death. I don't care about the particles making up my mind: I care about sustaining the never stopping information processing system which gives rise to my subjective experiance. It does not obviously follow that if my mind state were perfectly saved before I was shot in the head, and then at some distant point in the future a brain configured exactly like mine was created, that I would subjectively experience living on in the future. Not anymore than it makes sense to say that my recently deceased aunt lives on in my mother, her identical twin.