[Strilanc]

In Scott Aaronson's paraphrased words...

Quantum mechanics is just statistics, but operations preserve the 2-norm instead of the 1-norm. Instead of case weights (probabilities) adding up to 1, the squares of case weights (amplitudes) add up to 1. Everything else (the uncertainty principle, measurement mattering, no cloning, Bell inequalities, etc, etc, etc) follows.

[scythe]

One of my professors had a pithy way of saying this: "Quantum mechanics is the square root of classical mechanics."

[Koshkin]

Damn, now I need to understand the square root.

[codethief]

> Everything else (the uncertainty principle, measurement mattering, no cloning, Bell inequalities, etc, etc, etc) follows.

Did Aaronson provide a source for that?

[Strilanc]

He usually explains the details right after stating it. For example, you can follow along in his lecture notes: https://www.scottaaronson.com/democritus/lec9.html .

"Statistics but with the 2 norm" is really just a succinct way of stating the postulates of quantum mechanics, and obviously all effects of quantum mechanics are determined by the postulates. So you shouldn't really see this as a statement that's controversial at all.

[Kednicma]

In many cases, Wikipedia actually lists the analogy for classical probability theory. For several theorems, the proof for Hilbert spaces can be directly brought to the classical world. The main difference to keep in mind is that we have amplitudes, not probabilities, and they can be negative or complex. We recover typical probability theory by taking norms/squares of amplitudes.

The uncertainty principle follows via doing signal analysis on Fourier-transformed wavefunctions. A fun professional treatment from Baez et al. is [0], and 3blue1brown has an excellent two-video visual explanation [1][2]. The uncertainty principle turns out to be a special case of the sampling theorem (yes, that one! [3]), which itself turns out to be a special case of a result in sheaf theory [4].

Measurements matter because "observables don't commute"; taking linear operators on the complex numbers or other Hilbert spaces can have lasting effects which can't easily be undone. Combine this with "conservation of probability", which is formally mostly abstract nonsense [5], and we get mostly to Aaronson's point of view. (I would go further using the Free Will Theorem. [6])

When Aaronson says that the no-cloning principle is provable using probability, but for complex numbers, he's referring to the standard proof [7]. There are two connections to draw to typical probability theory. The first, and bigger, connection is that random variables can't be cloned in typical probability theory either! The second, deeper, connection is that Hilbert spaces give linear logics, which imply conservation laws for the information representing the particles to be cloned.

Finally, for the Bell inequalities, again there is a standard proof on Wikipedia [9] using typical probability theory. I'd like to mention the overlooked Kochen-Specker theorem [8], which forms the backbone of the Free Will Theorem [6]. Measuring a particle is like taking a sample of a random variable: We decide how we want to ask the particle, and the particle chooses a response that is both allowed by its probability distribution and also correctly represents its context.

[0] http://math.ucr.edu/home/baez/photon/schmoton.htm

[1] https://www.youtube.com/watch?v=spUNpyF58BY

[2] https://www.youtube.com/watch?v=MBnnXbOM5S4

[3] https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampli...

[4] https://arxiv.org/abs/1405.0324

[5] https://en.wikipedia.org/wiki/Probability_current

[6] https://en.wikipedia.org/wiki/Free_will_theorem

[7] https://en.wikipedia.org/wiki/No-cloning_theorem#Proof

[8] https://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem

[9] https://en.wikipedia.org/wiki/Bell%27s_theorem

[6nf]

Entanglement is a separate thing though I think.