Symmetry Structure of Mutually Unbiased Bases

Definitions and mathematical statement

For context, let $d,k\geq 1$ be integers. An orthonormal basis $B_a=(b_{a,1},\ldots,b_{a,d})$ of ${\mathbb{C}}^d$ satisfies $\langle b_{a,i},b_{a,j}\rangle=\delta_{ij}$. A family $(B_1,\ldots,B_k)$ is a family of mutually unbiased bases (MUBs) if \[ |\langle b_{a,i},b_{b,j}\rangle|^2=\frac{1}{d} \qquad (1\leq a,b\leq k,\ a\neq b,\ 1\leq i,j\leq d). \] These are contextual definitions; the supplied record does not specify values or intended quantifiers for $d$ and $k$, or a precise symmetry question about such families.

The recorded question is: "What is the symmetry in the MUB problem?" The supplied statement-verification field says that this is an expository heading in the report, immediately followed by identification of a wreath-product symmetry acting on the MUB problem, rather than a distinct unresolved claim. The record does not reproduce the factors of that wreath product, its action, or the mathematical object on which it acts. Consequently, no specific group action, classification problem, or conjecture can be stated from the supplied evidence alone.

The extracted text also contains a fragment asserting that, for $t=1$, the separate actions of $S$ on C[d]t and C[k]t are transitive, and that an "analytic d k decomposition" can be obtained from reference [5, Thrm. 3.1.1]. The symbols, spaces, decomposition, and reference are not defined in the supplied record. The leading "1For" may be an extraction artifact. This fragment is retained as an unresolved source qualification and is not promoted to a mathematical assertion with invented notation or hypotheses.

Thus the intended mathematical target is unrecoverable from this record. The supplied label open is a registry and dated-curation label, not a verified assertion that the expository question constitutes an open problem.

Short English statement

Mutually unbiased bases have squared overlap $1/d$ between vectors from different bases in ${\mathbb{C}}^d$. This record asks about their symmetry, but its supplied verification describes the question as an expository heading answered by a wreath-product action in the report. The action and intended target are missing, and an appended transitivity fragment has undefined notation. No precise open conjecture is recoverable without further source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004863$, title "Symmetry Structure of Mutually Unbiased Bases," exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The recorded proposed year is 2021; no individual proposer is supplied.
• Supplied report citation: Geometry and Optimization in Quantum Information (2022), Oberwolfach Reports, pp. 2665--2720, DOI 10.4171/owr/2021/49. Source URL and literature-source label "OWR report posing the symmetry question": https://doi.org/10.4171/owr/2021/49. This is the supplied report attribution; its contents have not been independently inspected here.
• Supplied literature source, labeled "Primary dimension-six MUB analysis": https://doi.org/10.1103/PhysRevA.83.062303. The record supplies no authors, title, or specific result connecting this reference to a precise symmetry target.
• Literature assessment dated exactly 2026-08-22: supplied status open. The supplied assessment says that the question lacks a precise mathematical target and that MUB constructions and constraints do not supply a general symmetry principle answering this record. This is supplied dated curation, not an independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field assigns statement_status: unrecoverable, describes the question as an expository heading followed by a wreath-product symmetry, and states that no separate clean statement is promoted. The extracted citation [5, Thrm. 3.1.1] cannot be identified from the supplied references.