>>11973091That's a pretty dicey way of stating how entanglement works. The point of nonlocal quantum games is that, after players have been given an entangled state, they can only act on their subsystem (particle) with local operations. That is, suppose we have two players who share an entangled state in . Then Alice, say, can only apply unitaries or projective measurements on her subsystem . If we write the state generically as where is a local basis for (and similarly for Bob), then given that Alice can only apply , we see Clearly Alice hasn't, and can't, affect Bob's subsystem.
My best guess is that when they say
>manipulating one entangled particle will effect the other one in the same waythey mean that the measurement statistics that Bob observes ARE affected by Alice's local operations. Indeed, you can show this by applying a POVM with and without and looking at distribution. So despite Alice not doing anything to Bob's particle, he observes different results. (Of course, depending on there may be certain symmetries such that specific leave Bob's observed distribution invariant in a given choice of basis.)
