Quantum does not contradict probability laws

Statistician Andrew Gelman writes:

Classical probability does not apply to quantum systems (causal inference edition) ...

If you recall your college physics, you’ll realize that the results of the two-slit experiment violate the laws of joint probability, ...

I discuss this in my linked blog post. But, in brief, the intuitive application of probability theory to the 2-slit experiment is that, if y is the position of the photon and x is the slit that the photon goes through, that p(y) = p(y|x=1)p(x=1) + p(y|x=2)p(x=2). But this is not true. As we all know, the superposition works not with the probabilities but with the probability amplitudes. Classical probabilities don’t have phases, hence you can just superimpose them via the familiar law of total probability. Quantum probabilities work differently.

This seems to be a widespread misconception. As Tim Maudlin explains in the comments, there is no contradiction with classical probability theory. In quantum mechanics, a photon is not a classical particle, but also has wave properties. The photon history is not just the sum of two particle possibilities. It can also be a wave that passes thru both slits at once.

The double slit experiment does show that light has wave properties. Everyone has agreed to that since 1803. If you deny that light is a wave that can go thru both slits at once, then you can get a contradiction. That is another way of saying the same thing. But the contradiction is with the classical particle theory of light, and not with probability theory.

There are people who have tried to make sense of quantum mechanics by using quantum logic or some modification to the laws of probability. These approaches have never worked.

I can't blame Gelman too much. There are a lot of physicists who, like Einstein, really want to believe that quantum mechanics is really a theory of imperfect info about hidden variables. It is not.

He argues:

Sure, a physical experiment can violate a mathematical law. The classic example is, if in a universe with closed curvature, you construct a large enough triangle, its angles will not add up to 180 degrees. Another classic example is that, for various particles, Boltzmann statistics do not apply, instead you have to use Fermi-Dirac or Bose-Einstein statistics. Boltzmann statistics is a mathematical probability model that does not apply in these settings. Another example is, in the two-slit experiment, p(A) does not equal the sum over B of p(A|B)p(B). In all these cases, you have a mathematical model that works (or approximately works) in some areas of application but not others. The math is not wrong but it does not apply to all settings.

This is silly. Yes, the math of flat space does not necessarily apply to curved space. Probability is a funny subject with multiple interpretations, but none of them are contradicted by light having wave properties.

He insists:

The 2-slit data indeed violate the laws of joint probability. I learned about this in physics class in college. In quantum mechanics, it is the complex functions that superimpose, not the probabilities. It is the application of the mathematics of wave mechanics to particles. The open question is whether it might make sense to apply wave mechanics to macroscopic measurements.

I would be interested in any textbooks say it wrong in this way.

Surely it must seem odd that we have a notion of probability that works in all situations except quantum mechanics, and we have some other notion that applies to quantum mechanics, but no one has figured out a way to make that probability notion apply to anything other than quantum mechanics. The answer is that quantum mechanics uses the same logic and probability that everyone else does.

Maudlin writes:

It ought to cause some pause that Feynman himself makes exactly this erroneous claim about the 2-slit experiment in the Lectures. Feynman does not mention locality, unitarity, or causality. He makes a straight claim about the data, based on a bad argument—exactly the argument I was attributing to Andrew. So if Feynman screwed this up, it would not be odd of many other physicists do too.

Feynman was a big advocate of particle interpretations of quantum mechanics. So he thought that the strangest part of quantum mechanics is the experiments showing wave behavior, like the double-slit experiment.