Ah,looks like I botched parts of this: > However, if you could somehow assign a complex amplitude to each state in a classical probability distribution, you wo

Ah,looks like I botched parts of this: > However, if you could somehow assign a complex amplitude to each state in a classical probability distribution, you would still be limited to manipulating only one amplitude at a time. This is probably just more confusing. What I should say is that classical probabilities have no physical manifestation that you can directly manipulate - they just denote our lack of information about a system. Amplitudes in quantum systems can be related to probabilities, but they don't represent lack of information. The probabilistic nature of quantum systems is deeper than that: measurements project superposition states onto classical states in a probabilistic way. This is For exponentially large superposition states, there are an exponential number of amplitudes. When we act on the state in certain ways, we update all of the amplitudes in parallel. There is no counterpart to this when acting on classical states, even when you have incomplete information about the state (and thus an exponentially large probability distribution). > But (especially) for the ones that do, I about the exponentially large set of amplitudes being manipulated in parallel. Let's try again.