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The probabilistic method uses arguments from probability to prove deterministic statements. This has been applied to diverse fields such as combinatorics, topology and number theory. In this method, you constructs a random variable based on the theorem that you want to prove. Using probabilistic arguments, you then show that the random variable has certain properties. From this the deterministic theorem follows.

I wonder if there is also a ‘quantum probabilistic method’. This would mean that you construct a (theoretical) quantum system based on a mathematical theorem that you want to prove. Then you prove that the quantum system must have certain properties, from which the mathematical theorem follows.

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    $\begingroup$ I guess you could argue that the recent proof that Connes' embedding conjecture is false uses such a method. $\endgroup$
    – Alubeixu
    Commented Apr 27 at 12:07
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    $\begingroup$ Here is the arXiv preprint of the paper that @Alubeixu referenced: arxiv.org/abs/2001.04383. I think this is a really good example for your question. $\endgroup$
    – Sam Hopkins
    Commented Apr 28 at 20:25

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The Hilbert-Polya approach to the Riemann hypothesis follows this path, by attempting to relate the zeroes of the Riemann zeta function to a quantum mechanical scattering problem. The probability distribution of the energy levels of a chaotic quantum system with broken time-reversal symmetry is conjectured to describe the local statistics of the zeta-function zeroes, see for example Physics of the Riemann Hypothesis.

Quantum algorithms provide a different area of quantum physics that is used to prove deterministic mathematical theorems. Drucker and De Wolf given an overview in Quantum Proofs for Classical Theorems. An example, is a proof of Jackson's inequality by means of a quantum probabilistic approach to polynomial approximation.

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  • $\begingroup$ It seems that this has not led to a proof. Are there also examples where applying quantum mechanics actually proved a theorem? $\endgroup$
    – Riemann
    Commented Apr 28 at 19:39
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    $\begingroup$ I have added one such example (and a reference which contains several more). $\endgroup$
    – Carlo Beenakker
    Commented Apr 28 at 20:19

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