Thread by @martinmbauer

Quantum mechanics can be demystified to some degree by realising that it's probability theory with a squared norm. E.g. you'd add classical probabilities (real numbers) p1, p2, .. < 1 with

p1+p2+...+ pn = 1

In QM instead you have amplitudes and

|w1|^2+|w2|^2+...+|wn|^2=1

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https://twitter.com/ns_nikhil9/status/1827945682935976236

For classical probabilities the linear norm is preserved by acting on P=(p1, ..., pn) with stochastic or Markov matrices M (non-neg matrices where all entries in a column add to 1)

M P = P' and p'1 + p'2 + ... +p2n =1

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In QM the quadratic norm is preserved by acting on W= (w1,...,wn) with unitary matrices U ∈ U(n)

U W = W' and |w'1|^2+|w'2|^2+...+|w'n|^2=1

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this alone shows how superpositions and interference are a thing in QM, but not for classical probabilities. If you act on a 2-state system 0 * Ia> + 1* |b> with the unitary matrix below you get 1/sqrt(2)( |a> + |b>)

In calculating the square the amplitudes interfere.

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In classical probability theory you only ever rescale the probabilities for a certain outcome, because stochastic matrices have non-negative entries and the norm isn't quadratic

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It also explains why amplitudes are intrinsically complex, whereas probabilities are real, because unitary matrices with complex entries preserve the 2-norm, but stochastic matrices can only ever have real entries

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Now one of my favourite observation is that if you look at higher power norms

|u1|^k + |u2|^k +... |un|^k = 1 for k>2 ,

you find that the only matrices that preserve these norms are simple permutations of the entries

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So in the space of k-norms nature has chosen by far the most interesting to construct QM

1-norm: stochastic matrices
2-norm: unitary matrices
k-norm (k>2) : permutation matrices

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