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Mutually Unbiased Bases in Dimension Six

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Construct a set of at least four mutually unbiased bases in dimension six, or prove that there are no seven mutually unbiased bases in $\mathcal{H}_6$.

Formulation references: [S1], [S2].

Short English statement

Construct a set of at least four mutually unbiased bases in dimension six, or prove that there are no seven mutually unbiased bases in $\mathcal{H}_6$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-32-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-32

The main theorem excludes seven arbitrary pairwise mutually unbiased orthonormal bases in C6. The problem statement is an either/or task: construct at least four or prove that seven cannot exist; the second alternative is proved in full. The interval-certificate argument claims exclusion of every four arbitrary mutually unbiased bases in C6 and combines it with the known three-base construction. This implies the problem statement alternative that no seven exist, with the stronger maximum N(6)=3. The claimed conclusion supplies the requested case and extends it in the directions just specified.

Mutually Unbiased Bases in Dimension Six

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Order-six Hadamard matrices outside the Tao class are claimed to satisfy the designated Fourier vanishing identities; exact certificates also exclude seven unbiased bases and imply an upper bound of five; The maximum MUB count in complex dimension six is claimed to be three, conditional on complete execution of the documented certified-arithmetic pipeline. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
The assertion excludes seven pairwise mutually unbiased orthonormal bases of $\mathbb C^6$. More explicitly, there are no bases $B_r=(b_{r,0},\ldots,b_{r,5})$, $1\le r\le7$, with \[ |\langle b_{r,i},b_{s,j}\rangle|^2=1/6 \qquad(r\ne s,\ 0\le i,j\le5). \] It does not by itself determine the maximum number below seven.
Result 2.
For pairwise mutually unbiased bases, $|\langle b,c\rangle|^2=1/6$ whenever $b,c$ belong to different bases of $\mathbb C^6$. The source reports a computer-assisted exclusion of four such bases and concludes \[ N(6)=3, \] where $N(d)$ is their maximal number in dimension $d$. The result is explicitly conditional on the appendix's arithmetic/compiler requirements and the reported complete execution; those computational requirements are not omitted from the claim.

Outline of the argument

Phase-invariant characters are rewritten as injective moments. Exact algebraic identities separate the Tao class, and integer/rational certificates verify the required moment inequalities; the supplied verifier and the completion theorem yield the stated MUB consequence.

An exhaustive seed cover is refined into certified phase balls. Completion estimates within each ball exclude two additional bases, and finite recorded execution covers all arbitrary complex configurations under the stated binary64 and compiler conditions.

Formal verification

For family 266, the selected statements give Fourier cancellation outside the Tao class and a bound of five on the MUB family, with a further Hadamard-ratio cancellation lemma. They do not establish the claimed maximum of three or certify the binary64 four-basis exclusion. The listed Comparator files are HadamardCubeFiber.lean, MUBSix.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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