Show that a given pair of MU bases in C6 does not extend to a triple (or quadruple) of MU bases.
Examples would include a proof that, for certain parameter values, {I, M6(1)}, {I, K6(2)} or {I, K6(3)} do not extend to a triple, in analogy with the isolated case {I, S6}.
The review's own fallback when the bigger problems are too hard: complete Table 2 by replacing numerical evidence of unextendibility with rigorous proof.
The review states the blocker in its own words, and it is a HARDWARE blocker: "For the symmetric, Hermitian and Szöllősi non-affine families, i.e. M6(1), B6(1) and X6(2), the available computational memory was insufficient to produce the relevant Gröbner basis, making certain approximations necessary. Thus, no rigorous conclusion regarding the existence of a fourth MU basis containing these families could be drawn." That is precisely the task our instrument performs — produce the basis, then replace the approximation with an exact certificate — and M6(1) is one of the three families Problem 10.6 names. A question abandoned for want of memory is an opportunity; a question abandoned because the mathematics resists is a wall. This one is on the record as the first kind.
Two risks, stated precisely. (1) The April 2025 invalidation of the family-level Chen–Yu chain touched M6(1) only — NOT the Karlsson families K6(2) or K6(3), which were never closed by that chain and are open for older reasons. An earlier version of this page said 'these families were reopened', which overstated it for two of the three. (2) The memory wall is a decade-old note in a review published four months ago; hardware has moved, and we will not be the only people who can read that sentence. If someone has already cleared it, the desk gate below should find them before we spend a core-hour.
- read Table 2 itself — which rows are numerical-only, and the citation behind each
- map every family name to the Życzkowski catalogue (the ten-minute check that would have killed the k=8 work)
- find the ORIGINAL memory-limited computation the review cites and establish what it actually ran out of, and on what hardware
- forward-citation sweep since the review (April 2026) for anyone who has since cleared it
Internal target: T1.4
Daniel McNulty, Stefan Weigert, Mutually Unbiased Bases in Composite Dimensions -- A Review
Quantum 10, 2051 (2026) · §10.2
https://doi.org/10.22331/q-2026-04-01-2051
Licence:
CC-BY-4.0
— the problem statement is quoted here under that
licence, with attribution, and has not been modified