Wait, I thought the superpower stuff only happens if you allow nonlinear transforms, not just nonunitary. Let’s add an additional restriction: let’s actually throw in some notion of locality, but even with the locality, abandon unitaryness. So our rules are “linear, local, invertable” (no rescaling aftarwards… not defining a norm to preserve in the first place)… or does locality necessitate unitarity? (is unitarity a word? Well, you know what I mean. Maybe I should say orthognality instead?)
Well, actually, also same question here I asked Eliezer. If you didn’t know squared amplitudes corresponded to probability of experiencing a state, would you still be able to derive “nonunitary operator → superpowers?”
Anyways, let’s turn it around again. Let’s say we didn’t know the Born rule, but we did already know some other way that all state vectors must evolve via a unitary operator.
So from there we may notice sum/integral of squared amplitude is conserved, and that by appropriate scaling, total squared amplitude = 1 always.
Looks like we may even notice that it happens to obey the axioms of probability. (it looks like the quanity in question does automatically do so, given only unitary transforms are allowed.)
Is the mere fact that the quantity does “just happen” to obey the axioms of probability, on its own, help us here? Would that at least help answer the “why” for the Born rule? I’d think it would be relevant, but, thinking about it, I don’t see any obvious way to go from there to “therefore it’s the probability we’ll experience something...”
Yep, my confusion is definately shuffled.
hrgflargh… (That’s the noise of frustrated curiousity. :D)
Stephen: I don’t have a postscript viewer.
Wait, I thought the superpower stuff only happens if you allow nonlinear transforms, not just nonunitary. Let’s add an additional restriction: let’s actually throw in some notion of locality, but even with the locality, abandon unitaryness. So our rules are “linear, local, invertable” (no rescaling aftarwards… not defining a norm to preserve in the first place)… or does locality necessitate unitarity? (is unitarity a word? Well, you know what I mean. Maybe I should say orthognality instead?)
Well, actually, also same question here I asked Eliezer. If you didn’t know squared amplitudes corresponded to probability of experiencing a state, would you still be able to derive “nonunitary operator → superpowers?”
Anyways, let’s turn it around again. Let’s say we didn’t know the Born rule, but we did already know some other way that all state vectors must evolve via a unitary operator.
So from there we may notice sum/integral of squared amplitude is conserved, and that by appropriate scaling, total squared amplitude = 1 always.
Looks like we may even notice that it happens to obey the axioms of probability. (it looks like the quanity in question does automatically do so, given only unitary transforms are allowed.)
Is the mere fact that the quantity does “just happen” to obey the axioms of probability, on its own, help us here? Would that at least help answer the “why” for the Born rule? I’d think it would be relevant, but, thinking about it, I don’t see any obvious way to go from there to “therefore it’s the probability we’ll experience something...”
Yep, my confusion is definately shuffled.
hrgflargh… (That’s the noise of frustrated curiousity. :D)