Quantum mechanics can be construed as a way to limit computation and avoid arithmetic overflows.
You’re missing JoshuaZ’s point. Quantum mechanics at first looks like it avoids arithmetic overflows, by letting you get away with finite precision. But it doesn’t, really. There’s still arbitrarily-precise numbers that the universe “keeps track of”—but they’re amplitudes, not observables.
In quantum mechanics, amplitudes are complex numbers, that seem to be treated as mathematically exact, out to indefinitely much precision. And the possibility of quantum computing suggests that this isn’t a purely an artifact of our models -- you can potentially do useful computation using the low-order bits of the amplitudes.
Question for the physics types here: If amplitudes didn’t add precisely linearly, could we tell? Is there a sensitive test for linearity?
The notion of “a single particle” turns out to be problematic. The thing that requires arbitrarily much information is the amplitude for a particle to be in some particular state.
I should emphasize that amplitudes aren’t just a creature of our models—they are the thing that interfere to give you an diffraction pattern, or the shapes of an electron orbital cloud, or that get manipulated in a quantum computation.
You’re missing JoshuaZ’s point. Quantum mechanics at first looks like it avoids arithmetic overflows, by letting you get away with finite precision. But it doesn’t, really. There’s still arbitrarily-precise numbers that the universe “keeps track of”—but they’re amplitudes, not observables.
In quantum mechanics, amplitudes are complex numbers, that seem to be treated as mathematically exact, out to indefinitely much precision. And the possibility of quantum computing suggests that this isn’t a purely an artifact of our models -- you can potentially do useful computation using the low-order bits of the amplitudes.
Question for the physics types here: If amplitudes didn’t add precisely linearly, could we tell? Is there a sensitive test for linearity?
So even in quantum mechanics, it takes an infinite amount of information to represent a single particle? That’s a problem.
It’s a problem for us. But the universe doesn’t have to care.
The notion of “a single particle” turns out to be problematic. The thing that requires arbitrarily much information is the amplitude for a particle to be in some particular state.
I should emphasize that amplitudes aren’t just a creature of our models—they are the thing that interfere to give you an diffraction pattern, or the shapes of an electron orbital cloud, or that get manipulated in a quantum computation.