The maximum number of mutually unbiased bases in dimension six

Introduction

Two orthonormal bases $\mathcal{B}$ and $\mathcal{C}$ of $\mathbb{C}^{d}$ are mutually unbiased if

Here $\langle u,v\rangle=\sum_j\overline{u_j}v_j$. For $d\ge2$, let $N(d)$ be the largest number of pairwise mutually unbiased orthonormal bases in $\mathbb{C}^{d}$. Our main result is the following.

Theorem 1.1 (Computer-assisted). Under the arithmetic and compiler conditions in Appendix A, the fresh complete execution documented in Section 7.5 certifies that there are no four pairwise mutually unbiased orthonormal bases in $\mathbb{C}^{6}$. Consequently, $N(6)=3$.

A common unitary transformation makes one basis the coordinate basis. The second is then the column basis of $H/\sqrt{6}$, where $H$ is a complex matrix satisfying

Such an unscaled matrix is called a complex Hadamard matrix. The upper-bound problem is to exclude an extension of this pair by two further bases, for every possible $H$. The second basis is unrestricted: the argument covers arbitrary complex Hadamard matrices, including those outside previously studied families.

Historical context and scope

Mutually unbiased bases describe projective measurements for which a state from one basis produces the uniform outcome distribution in every other basis. This finite-dimensional form of complementarity appears in Schwinger’s unitary-operator framework [22]. Ivanović developed the state-determination problem and prime-dimensional constructions [13]; Wootters and Fields proved the statistical optimality of complete sets for state determination and constructed them in every prime-power dimension [24]. One has $N(d) \le d+1$; an operator-space proof and further prime-power constructions are given in [1]. Dimension six is the first dimension outside the prime-power class.

The tensor-product construction gives three bases in $\mathbb{C}^{6}$ [16], Section 4. Zauner’s dimension-six MUB conjecture asserts that this lower bound is sharp. Its origin is traced to his 1991 diploma thesis [25]; see also the historical account in [20], Section 2.3. His doctoral thesis states the corresponding quantum-design conjecture [26], Section 3.3, later translated in [27]. The equation $N(6)=3$ appears explicitly as Conjecture 1 of [16], Section 4.

Several complementary approaches developed the dimension-six problem. Grassl proved that the standard–Fourier pair cannot belong to four mutually unbiased bases [9], Theorem 2. Bengtsson and collaborators studied its unbiased vectors and the surrounding landscape of Hadamard families [2]. Numerical searches by Butterley and Hall [6] and by Brierley and Weigert [4] supplied evidence for the conjecture. Brierley and Weigert also used algebraic equations to enumerate unbiased vectors for specified Hadamards [5]; their Section 3.4 supplies rigorous approximation bounds for exact input matrices, while their Section 4.3 explains the limitations introduced by rounding coefficients in sampled non-affine families.

The closest antecedent of the present proof is the certified discretization of Jaming, Matolcsi, Móra, Szöllősi and Weiner. They proved that the standard basis together with any member of the two-parameter Fourier family cannot be extended to four bases [15], Theorem 1.4. Their Section 3 combines phase intervals, rigorous error bounds, subdivision, and exhaustive compatibility tests. Jaming, Matolcsi and Móra subsequently formulated an unrestricted discretization program [14], Sections 2–4. The present proof completes an exhaustive cover of arbitrary second bases. The completion and displacement estimates described below reduce this cover to a finite search with certified error bounds.

Recent work on triples retains conjectural sufficient conditions for $N(6)=3$ [19], Section 4. The classification preprint of Cárdenas Wuttig and Tindall concerns individual order-six Hadamard matrices and leaves their MUB compatibility unresolved [7], Section VI. Our proof constructs a cover of arbitrary second bases directly. Its upper bound combines that exhaustive cover with local exclusion of two further bases and the complete execution and arithmetic contract of Theorem 7.1.

The covering and exclusion strategy

The second basis varies throughout the search, so the cover must retain its possible displacement while testing additional vectors. The proof first covers every second basis by balls in its real entrywise phases, then excludes two further bases uniformly throughout each ball. Figure 1 summarizes the five parts of this argument.

1. Row and column rephasings make the first row and column of $H$ equal to one. The first two rows and columns then contain nine remaining phases. A symmetry reduction and necessary interval tests cover their possible values by finitely many seed balls (Section 2).

2. In Stage A, write $H$ in blocks with row and column sizes $2+4$. The seed fixes the upper-left, upper-right, and lower-left blocks up to their phase displacements; the lower-right 4-by-4 block remains to be completed. At each seed center that passes the frame checks, the upper-right block and the conjugate transpose of the lower-left block have rank two, and hence two-dimensional kernels in $\mathbb{C}^{4}$. Compressing the unknown block between these kernels gives a 2-by-2 matrix. The Hadamard equations bound its distance from $\sqrt{6}$ times its unitary polar factor. A closed cover of $U(2)$, the group of 2-by-2 unitary matrices, therefore accounts for every feasible completion. Uniform Taylor bounds either exclude a cell or place all its exact completions in certified phase balls (Sections 3 and 4). The frames for these kernels remain fixed at the seed center, which need not itself be feasible.

3. In Stage B, each ball carries numerical directions approximating the tangent directions of the Hadamard equations. A verified residual identity bounds the error in representing every exact solution displacement by these directions. Covering the tangent coordinates and retaining this normal bound yields smaller balls covering all the same exact solutions (Section 5). No manifold parametrization is required.

4. In Stage C, a phase-bin search in each final ball covers every vector unbiased to the first two bases by small vector balls. Make these vector balls the vertices of two graphs: an $\mathcal{O}$ edge retains a possible orthogonal pair, and a $\mathcal{U}$ edge retains a possible unbiased pair. Necessary tests remove only impossible edges. Two additional bases would require two disjoint six-cliques in $\mathcal{O}$, with all 36 cross edges in $\mathcal{U}$. The finite search excludes this pattern (Section 6).

5. The complete execution accounts for every seed, every refined ball, and every initially unresolved record. The fallback finishes with no unresolved record (Section 7).

The distinction between proposing a numerical object and certifying it is used throughout. A floating-point center need not be an exact Hadamard matrix. A computed small eigenspace need not parametrize a component. A numerical linear solve need not be accurate: its output is accepted only when a separate bound verifies the required inequality. Every rejection uses a necessary condition with an explicit safety margin. A failed certification or an exhausted search cap retains an unresolved region; it cannot prove nonexistence.

All interval cells may be taken closed. Their overlap, repeated phase representatives, and redundant balls or graph vertices can weaken the exclusion but do not omit a possible configuration. The arithmetic premises are stated, and the finite ranges and rounding margins are proved, in Section A. The accompanying execution and build archives, identified in verification/computation/README.md, preserve the evidence for the fresh run documented in Section 7.5. This proof uses that run. The separate September 7 execution reported in the source manuscript has not been authenticated here.

The elementary lower bound

The lower bound uses only the tensor-product construction mentioned above. We give it explicitly to separate attainment of three bases from the computer-assisted exclusion of four.

Lemma 1.2 (Three mutually unbiased bases in dimension six). There exist three pairwise mutually unbiased orthonormal bases in $\mathbb{C}^{6}$.

Proof. Identify $\mathbb{C}^{6}$ with $\mathbb{C}^{2} \otimes\mathbb{C}^{3}$ in the standard product coordinates. Define

These matrices are unitary, and for $r\ne s$ every entry of $A_{r}^{*}A_{s}$ has squared modulus $1/2$; the only cross product that is not immediate from the entries is

Let $\omega=e^{2\pi i/3}$, and set

The identity $1+\omega+\omega^{2}=0$ shows that $F$ is unitary; $D$ is unitary as well. Both $F$ and $DF$ have entries of squared modulus $1/3$. For their cross product,

If $t=k-j$ is read modulo 3, the sum on the right is $1+2\omega$ for $t=0$ and $2+\omega^2$ for $t=1,2$. Each has squared modulus 3. Consequently every entry of $B_r^*B_s$ has squared modulus $1/3$ when $r\ne s$.

For $r=0,1,2$, let the columns of $U_r=A_r\otimes B_r$ form the $r$th basis. Each $U_r$ is unitary. For $r\ne s$,

so every entry has squared modulus $(1/2)(1/3)=1/6$. Thus these three bases are mutually unbiased, and $U_0=I_6$.

A Lie-algebra consequence

An orthonormal basis $\mathcal{B}$ of $\mathbb{C}^6$ determines the Cartan subalgebra $\mathfrak{h}_{\mathcal{B}}$ of traceless matrices diagonal in that basis. It is spanned by the matrices $bb^*-I/6$ for $b\in\mathcal{B}$. The Killing form on $\mathfrak{sl}_6(\mathbb{C})$ is $K(X,Y)=12\operatorname{tr}(XY)$; two subalgebras are Killing-orthogonal when this form vanishes between them. For unit vectors $b,c$,

Thus two bases are mutually unbiased exactly when their associated Cartan subalgebras are Killing-orthogonal. These algebras are closed under the standard adjoint, and the converse correspondence uses this same adjoint condition.

Corollary 1.3 (Adjoint-closed Cartan subalgebras). Under the arithmetic and compiler conditions of Appendix A, using the complete execution documented in Section 7.5 as in Theorem 1.1, the maximum number of pairwise Killing-orthogonal Cartan subalgebras of $\mathfrak{sl}_6(\mathbb{C})$ that are all closed under the same standard adjoint $X\mapsto X^*=\overline{X}^{T}$ is three.

Proof. By [3], such collections of Cartan subalgebras correspond to collections of mutually unbiased orthonormal bases of the same cardinality. Theorem 1.1 gives the upper bound under its stated contract, and Lemma 1.2 gives attainment.

Equivalences and an exhaustive seed cover

The goal is to replace every possible second basis by one of finitely many nine-phase seed balls. Closed phase discretization and necessary compatibility tests follow the approach of [15] and [14]. The symmetry reduction and shared-coordinate estimates below prove completeness of this particular seed cover.

We use phases in cycles and write

with entrywise application to vectors and matrices. Matrix indices run from 0 to 5, and a phase matrix is stored in row-major order, with $(i,j)$ in position $6i+j$. The Hermitian inner product is $\langle u,v\rangle=\sum_j\overline{u_j}v_j$, and $A^*$ denotes conjugate transpose. Vectors have their Euclidean norms; a matrix regarded as a phase or displacement vector has Frobenius norm $\lVert\cdot\rVert_F$. The notation $\lVert\cdot\rVert_{\operatorname{op}}$ denotes the induced Euclidean operator norm. We identify a complex equation with its real and imaginary parts, in that order, so that a complex scalar and its realisation have the same norm. An angle is in radians only when this is stated explicitly.

Equivalences and phase gauges

Definition 2.1. A matrix $H \in\mathbb{C}^{6 \times6}$ is a complex Hadamard matrix here if every entry has modulus one and $H H^{*} = 6I$. It is extendible if the standard basis and the columns of $H/\sqrt{6}$ belong to a family of four mutually unbiased orthonormal bases.

A common unitary maps the first member of any proposed family to the standard basis. Unbiasedness to that basis forces the entries of the second basis matrix to have modulus $1/\sqrt{6}$. Thus it is $H/\sqrt{6}$ for a matrix as in Definition 2.1; since $H$ is square, $H H^{*} = 6I$ also gives $H^{*}H = 6I$. Every such $H$ admits a real phase lift $P$ with $H = e(P)$.

Lemma 2.2 (Equivalences). Extendibility is invariant under row or column permutations, multiplication of rows or columns by unit scalars, complex conjugation, and transposition.

Proof. Permuting or rephasing columns only changes the order or phases of vectors within the second basis. A row permutation or row rephasing is a common unitary operation on all four bases; it preserves the standard basis up to the same irrelevant changes. Complex conjugation preserves both orthogonality and the moduli of inner products.

For transposition, put $U = H/\sqrt{6}$. Apply the common unitary $U^{*}$ to a family whose first two basis matrices are $I,U$. These two matrices become $U^{*},I$. Interchange their roles and conjugate every coordinate. The first two matrices are now $I,U^{T}$. All these operations are reversible, which proves the assertion in both directions.

The transposition argument applies to an exact Hadamard matrix. It does not assert that an approximate center supplies a unitary change of basis.

For a real $6$-by-$6$ matrix $z$, define double centering by

Its range is the horizontal subspace of matrices with every row sum and every column sum zero. We also use unwrapped dephasing,

Lemma 2.3 (Horizontal phase representatives). The map $\Pi$ is the orthogonal projection onto the horizontal subspace. In particular, $\lVert\Pi z\rVert_F \le\lVert z\rVert_F$. If two phase representatives have difference $z$, the second is equivalent under row and column rephasings to the first plus $\Pi z$. This statement remains valid after arbitrary entrywise integer changes of phase representatives.