not sure that we're talking about the same. My understanding here is that one calculates probabilities the same way. It is the interpretation different. In Copenhagen the particle is supposed to carry all probabilities which magically "collapse" on the measurement, while in statistical a given particle is in "collapsed" state to start with (there is a distribution of the states over the ensemble), and the probabilistic model describes the evolution of these states and distribution of final states - there is no collapse on measurement, we just measure specific state.
>I'm unclear why you're bringing up Hitach's experiment and what you think it shows. Maybe you could elaborate? My understanding is that the raison d'être of the experiment is to show that multiple particles are not necessary for quantum interference to occur, and that a single particle is perfectly capable of destructively interfering with itself.
in Hitachi experiment it is shown explicitly clear that quantum interference is emerging only for multiple particles. It is basically a visualization of underlying probability distribution, like it happens in any statistical experiment when enough samples are taken. During the Hitachi experiment nothing happened that looks like or requires an explanation by a single particle supposedly interferencing with itself.
If you don't see this in Hitachi experiment, lets run the following experiment. Imagine 2 doors in a wall, and imagine another, parallel, wall at several meters distance from the doors. Imagine that frogs jump out, one at a time, from either door pretty randomly. The frogs jump in general direction of the wall opposite the doors. The precise direction of each frog is varying a bit. It takes a frog several jumps to reach the wall. The probability density of the places where the frog's legs touch the ground is square of cos(pi*x/A) (correctly scaled to be a probability density) where x is the distance from the door the frog jumped out, and A is the avg. frog jump length. A frog only exist (i.e. can interact with anything else) when it touches the ground, and doesn't exist when it is flying during the jump. Whenever a frog lands near the wall, say not farther than a body distance from the wall, it touches the wall and leaves a wet spot.
It is easy to see that after a big enough bunch of frogs, there would be an "interference pattern" of dry and really wet areas on the wall - i.e. some places have low probability of a frog landing near it while some have high, and that probability distribution looks like an "interference pattern". No superposition, no destructive interference of a frog with itself is necessary to observe the effect. This is what Hitachi experiment shows.
>This is pretty direct evidence for superposition,
yes, an interference of a particle with itself would be such evidence. I'm yet to learn about an experiment which can only be explained by such interference.
> Any successful theory of physics must explain how 15 was factored using Shor's Algorithm.
agree. While 15 seems too small a number to exclusively lock a superposition explanation, i don't have another ready. I'd like to have quantum computers and other quantum miracles too :) I just want them to be a bona fide miracles, not figments of our interpretation :)