Exactly three mutually unbiased bases in dimension six

The explanation

Start with two directions on a flat page, straight across and straight up. In quantum physics, a pair like this can stand for a measurement with two outcomes. Now tilt both by 45 degrees. Each tilted direction makes the same angle with each original one. Prepare a state along one set and measure with the other, and both outcomes are equally likely, one half each. Two such sets are called mutually unbiased bases.

In two dimensions you can fit three sets that are all unbiased with each other. The third uses complex numbers: the directions (1, i) and (1, −i), scaled down by √2. A check of every pair of vectors from different sets gives exactly one half each time. In d dimensions, at most d + 1 such sets can exist. In 1989, William Wootters and Brian Fields built d + 1 sets whenever d is a prime or a power of one.

Six is the first dimension that is neither. Combining the three sets for dimension 2 with three sets for dimension 3 gives three sets for dimension 6. A computer check confirms that every pair of vectors from different sets there gives 1/6, as it should. Can a fourth set be added? Gerhard Zauner conjectured in his 1991 thesis that it cannot, so the answer is three. Numerical searches agreed, but searches cannot cover every case. Earlier proofs excluded a fourth set only for special families of second bases.

The single difficulty is that the second set can be anything. After a change of view, the first set is the plain axes. The second set becomes a 6 by 6 table of complex numbers, all of size one. That table can vary in many directions at once, including ones outside every studied familySource 1. So a proof must rule out two more sets for every possible second set, not only the known ones.

This paper claims to do that with a computer-assisted search. It covers every possible second set by a finite list of small balls. For each ball, it bounds how far any true solution inside can move. Then it searches for vectors unbiased to the first two sets. It shows that no two further sets can be built from them anywhere in the ball. Every rejection uses a test with an explicit safety margin, so rounding cannot hide a real solutionSource 2. The search finished with no region left unresolved.

That makes the certainty statement unusual. The result holds under stated conditions on the computer arithmetic and compiler, for the complete run the paper documentsSource 3. A companion paper takes a different route with exact whole-number arithmetic. It rules out seven setsSource 4. With an earlier theorem of Mihály Weiner, that gives at most five. It also settles a related conjecture of Matolcsi, Ruzsa and Weiner about certain sums over these tablesSource 4.

How sure is this? A computer proof checker called Lean has checked formal versions of the bound of five and of that related conjecture. It has not checked the main claim that three is the maximum. That claim rests on the documented computer search, and no specialist review has been published.

What it doesn't show. This does not settle the maximum number of such sets in any other dimension that is not a prime power.