Three complementary measurements in dimension six—and why a fourth is excluded

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Family 266 · Mathematical physics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

Some quantum measurements are maximally incompatible in a precise sense: knowing the outcome of one leaves every outcome of another equally likely. How many such measurements can coexist in a six-dimensional quantum system?

Represent each measurement by an orthonormal basis of $\mathbb C^6$. Two bases are mutually unbiased when every cross-basis squared inner product equals $1/6$. Three pairwise mutually unbiased bases are known to exist. The long-standing question is whether a fourth can exist.

M1 claims that the maximum is exactly three, through a computer-assisted exclusion covering arbitrary complex bases under specified arithmetic and compiler conditions. M2 proves a different structural claim about Fourier sums of complex Hadamard matrices and gives an exact-arithmetic obstruction to a complete seven-basis family. The two conclusions should remain distinct: M2’s basis-count consequence is an upper bound of five, not three.

2. Background and essential definitions

In $\mathbb C^d$, let $B=\{b_1,\ldots,b_d\}$ and $C=\{c_1,\ldots,c_d\}$ be orthonormal bases. They are mutually unbiased if

$$|\langle b_i,c_j\rangle|^2=\frac1d \qquad(1\leq i,j\leq d).$$

The quantum interpretation follows directly from the rule for measurement probabilities. If a state is one vector of $B$, measurement in $C$ gives a uniform distribution. “Unbiased” here is a relation between two whole bases, rather than a statement about statistical estimation bias.

Write $N(d)$ for the largest number of pairwise mutually unbiased bases. The general upper bound is $d+1$. Complete families reaching that bound exist when $d$ is a power of a prime. Six is the smallest dimension outside that construction, since $6=2\cdot3$.

A common unitary change of coordinates can make one basis the standard coordinate basis. Every vector in a second unbiased basis must then have coordinates of modulus $1/\sqrt d$. Writing its vectors as columns gives a normalized complex Hadamard matrix $H/\sqrt d$, where

$$|H_{ij}|=1,\qquad HH^*=dI.$$

The entries may be arbitrary complex phases. Restricting them to a finite set of roots of unity would change the problem.

Multiplying rows or columns by phases, or permuting them, preserves the relevant Hadamard equivalence. One may make the first row and first column all ones; this is called dephasing. It removes redundant coordinates without claiming that all remaining matrices belong to a known parametrized family.

For M2, a balanced integer vector $a=(a_0,\ldots,a_5)$, with $\sum_i a_i=0$, defines a character sum

$$g_H(a)=\frac16\sum_{k=0}^5\prod_{i=0}^5H_{ik}^{a_i}.$$

It averages a phase monomial over the columns. Balance makes it invariant under column rephasing. These sums are Fourier coordinates on the torus of possible phases, and orthogonality imposes relations among them.

3. Development history and previous partial results

Mutually unbiased bases arose in the mathematical description of complementary quantum measurements. Schwinger’s unitary-operator framework, Ivanović’s state-determination work, and Wootters–Fields constructions connected them to quantum tomography and prime-power dimensions. The general $d+1$ bound and its prime-power attainability leave composite dimensions with a different arithmetic structure.

The conjecture $N(6)=3$ is associated with Zauner. The lower bound follows from tensor products: take three mutually unbiased bases in dimension two and three chosen bases in dimension three, and pair corresponding factors. For cross-basis tensor vectors, squared overlaps multiply to $(1/2)(1/3)=1/6$.

The upper bound resisted several approaches. Grassl excluded four bases containing the standard–Fourier pair. Numerical searches and algebraic studies examined many Hadamard families, but failure to find a fourth basis is not a universal exclusion.

A close predecessor is Jaming, Matolcsi, Móra, Szöllősi and Weiner’s certified Fourier-family analysis. It ruled out extending the standard basis together with any member of a specified two-parameter Fourier family to four bases. Their method combined phase intervals, error bounds, subdivision and exhaustive compatibility tests. Jaming, Matolcsi and Móra subsequently outlined an unrestricted discretization program.

A separate line uses Fourier and combinatorial identities. Matolcsi, Ruzsa and Weiner conjectured a particular degree-six character cancellation outside one exceptional cubic Hadamard class. Maxwell and Brierley established that cancellation for the Karlsson family. M2 claims to remove the family restriction and adds a finite positivity certificate for the complete-family problem.

Thus the two manuscripts extend different partial solutions: exhaustive covering beyond previously specified second bases, and universal Fourier identities beyond previously specified Hadamard families.

4. What the new manuscripts claim

Exactly three bases, with a computational contract

M1 states that, under its binary64 arithmetic and compiler assumptions, the complete execution documented in its section 7.5 excludes four arbitrary pairwise mutually unbiased bases in $\mathbb C^6$. Combined with the elementary construction of three, it claims

$$N(6)=3.$$

The second basis is unrestricted. The method covers all complex Hadamard matrices of order six rather than relying on a classification into familiar families.

The computational premises are explicit. They include the stated floating-point rounding model, separately evaluated operations, specified square-root behavior, and compiler conditions excluding transformations incompatible with the error analysis. The paper reports a complete execution accounting for initially unresolved records. These are descriptions of the manuscript’s result and evidence; they are not a separate execution or verification performed for this digest.

Universal Fourier vanishing

Let

$$\alpha=(1,1,1,-1,-1,-1).$$

M2 claims that every order-six complex Hadamard matrix outside Tao’s cubic equivalence class satisfies

$$g_H(\pi\alpha)=0$$

for every coordinate permutation $\pi$. The exceptional class has entries that, after the allowed equivalences, are cube roots of unity. The manuscript notes that its displayed representative also appeared in Moorhouse’s earlier work.

A related structural step says that if both $H$ and its entrywise square are Hadamard, then $H$ belongs to that cubic class. Entrywise squaring means squaring each entry, not matrix multiplication $H^2$. This classification is used to identify the exception in the Fourier argument.

M2 also claims there is no complete family of seven mutually unbiased bases in dimension six. It then invokes Weiner’s completion theorem: any $d$ mutually unbiased bases in $\mathbb C^d$ extend to a complete $d+1$-basis family. Excluding seven therefore excludes six as well, giving

$$N(6)\leq5.$$

Its certificates use integer and rational arithmetic, and it includes a verifier. This structural and exact-arithmetic contribution is meaningful independently of M1’s sharper count.

5. Why the advance matters

Provisional importance: 4/5. The six-dimensional MUB problem is a prominent finite-dimensional question in quantum information and discrete mathematics. Six is the first place where prime-power constructions do not settle the maximal count, so a complete exclusion of four bases would mark a major boundary of those constructions.

The Fourier result has its own value. It constrains every individual order-six Hadamard matrix, including matrices not assumed to participate in any large unbiased family. Such structural identities can support further analysis of complementary measurements and complex Hadamard geometry.

The rating reflects a major resolution within a focused dimension and problem class. It does not imply a classification of all composite dimensions or of all triples in dimension six. The two papers also illustrate complementary ways of obtaining a universal conclusion: a finite covering with controlled numerical error and exact identities derived from positivity.

6. The main idea

Cover every possible second basis

M1 first removes phase and permutation redundancies. A small initial block leaves nine phases to cover. Necessary interval tests remove impossible seed regions while retaining all feasible choices.

For a surviving seed, the remaining block must satisfy both unitary and fixed-modulus conditions. The argument compresses part of the unknown completion to a two-dimensional unitary problem. A closed cover of $U(2)$, together with Taylor remainder bounds, either excludes a cell or places its possible completions inside certified phase balls.

The balls are refined using approximate tangent directions to the Hadamard equations plus a bound on normal displacement. The numerical directions need not parametrize a smooth manifold. Their role is only to give a controlled cover of every exact solution still possible.

Inside each final ball, the method covers vectors unbiased to the first two bases. These vector balls become vertices in two graphs. One graph retains pairs that might be orthogonal; the other retains pairs that might be unbiased. Two further bases would require two disjoint six-vertex cliques in the orthogonality graph, with all cross-pairs connected in the unbiasedness graph.

The claimed execution excludes this pattern everywhere. A failed numerical certification or an exhausted search cap is retained as unresolved, rather than counted as a contradiction. The logical distinction is between proposing a useful center and certifying an exclusion for the whole region around it.

Derive an exact positivity contradiction

M2 averages monomials over distinct row and column indices. Orthogonality supplies linear relations among these averages. Products of character sums can be expanded into the same moment coordinates by tracking which sampled columns coincide.

The first certificate expresses an average of nonnegative quantities as zero. Each summand must therefore vanish. This forces a dichotomy: either the desired character sums vanish, or the entrywise square of the matrix is also Hadamard. Algebraic identities for row ratios then identify the second alternative with the cubic class.

For a complete unbiased family, further moments involve several transition matrices and shared columns. Explicit positive quadratic forms would have a combined value forced to be negative by their exact coefficient identities and the bound that every normalized moment has modulus at most one. That contradicts positivity.

The certificate coefficients are rational, but the actual Hadamard entries need not be. Exact arithmetic verifies identities valid for arbitrary complex phases. This is why the approach is not restricted to a rational or root-of-unity sample of the original problem.

7. Scope, limitations, and open questions

M1’s strongest claim retains its stated arithmetic and complete-execution conditions. M2’s rational certificates establish the Fourier theorem and the weaker upper bound described above. They should not be conflated into an assertion that the exact Fourier calculation alone proves $N(6)=3$.

The family does not classify every triple of mutually unbiased bases, every order-six complex Hadamard matrix, or the maximum number of bases in all non-prime-power dimensions. Nor does nonexistence of an exact fourth basis quantify how close an approximate fourth basis can come to satisfying the conditions.

Questions about approximate complementarity, structural classifications of triples, and analogous obstructions in other composite dimensions remain separate. The source statements are reported at their stated scope, including the computational premises of the sharper theorem.

8. Sources and further reading

The primary links in section 3 lead to the Fourier-family exclusion, the unrestricted discretization proposal, and the Fourier/combinatorial approach. M1 explains the exhaustive covering and the contract for its reported computation. M2 gives the individual-matrix Fourier theorem, the cubic exception, the complete-family certificate and the use of Weiner’s completion theorem. Its appendix contains the exact verifier as part of the source exposition.

Source manuscripts

• M1. OpenAI, The maximum number of mutually unbiased bases in dimension six (September 24, 2026).
• M2. OpenAI, Exact Fourier certificates for complex Hadamard matrices of order six (September 24, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.