Doing induction like a physicist

If something is true for dimension 2, it doesn’t mean much. We know that 2 is very special. The set of valid quantum states is a sphere, we can have a basis of unitary and Hermitian matrices for the Hilbert space, extremal quantum correlations can always be produced by projective measurements, you can have a noncontextual hidden variable model, and so on. None of that holds for larger dimensions.

If something is true for dimensions 2 and 3, that’s already much better, but by no means conclusive. There exists a simple formula for SIC-POVMs for these dimensions, that doesn’t work for 4 onwards. If we go beyond dimensions, there are more interesting examples: for 2 qubits, there exists a single class of entangled states, namely the $|00\rangle+|11\rangle$ class. For 3 qubits, there are two classes, namely the $|000\rangle + |111\rangle$ and $|001\rangle + |010\rangle + |100\rangle$ classes. One could hope that for 4 qubits we would have three classes, but no, there are infinitely many. The same non-pattern happens for Bell inequalities. For bipartite inequalities with two outcomes per party, if each has 2 settings then there exists only one facet inequality, the CHSH. If each party has 3 settings, then there are two facets, CHSH and I3322. If they have 4 settings, though, there are 175 different facets. Ditto if you fix the number of settings to be 2, and increase the number of outcomes. For 2 outcomes, again only CHSH, for 3 outcomes you have CHSH and CGLMP, and for 4 outcomes many more.

If something is true for dimensions 2, 3, and 4, then it will be also true for dimension 5, so we skip this one.

If something is true for dimensions 2, 3, 4, and 5, it is very good evidence that it will be true for all dimensions, but it is still not enough for a proof by physicist induction. We have even primes, odd primes, and prime powers, but no non-trivial composite numbers. MUBs are a good example, they exist for dimensions 2, 3, 4, and 5, but not 61.

If something is true for dimensions 2, 3, 4, 5, and 6, that’s it. It will be true for all dimensions. SIC-POVMs are a good example. It is not too hard to construct analytical examples for dimensions 2, 3, 4, 5, and 6, and from that we know that they always exist2.

This is of course not true in mathematics, which is a demanding and capricious mistress. The most horrifying example I know is the logarithmic integral. Quantum mechanics, on the other hand, is a mother. She will not humiliate you, she will not lead you astray. She only wants you to do a bit of honest work with small dimensions, and she will reward you with the truth.

The only potential counterexample I know is the Tsirelson bound of the I3322 inequality, which is supposed to be 0.85 for dimensions 2 to 8, and from dimension 9 onwards it starts increasing. I don’t count it as an actual counterexample because nobody managed to actually prove that the Tsirelson bound is 0.85 for dimensions 2 to 6, there is just numerical evidence. And I do demand a proof for this part of physicist induction, the reasoning is already flimsy enough as it is.

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