This manuscript provides a complete algebraic proof excluding the μ_5 branch—the most challenging generic branch in the longstanding open problem of constructing a third mutually unbiased basis (MUB) in dimension d=6 (related to Zauner's conjecture). The proof introduces a novel algebraic framework based on the Schur-closure identity of the Gram matrix, yielding a fundamental factorized master equality. Through rigorous phase-alignment reduction and subresultant elimination, we demonstrate that the regular physical regime inevitably forces a "spectral collapse". This mechanism mathematically annihilates the Vandermonde determinant, strictly driving the physical phase vector to the trivial zero vector and yielding an absolute contradiction. Furthermore, we provide explicit Cramer coefficients, boundary-lifting lemmas, and a self-contained numerical witness certificate to ensure full mathematical traceability. Our results definitively show that the μ_5 branch carries no non-degenerate physical solution, establishing a major milestone toward the full algebraic resolution of the d=6 MUB problem.
This manuscript provides a complete algebraic proof excluding the μ_5 branch—the most challenging generic branch in the longstanding open problem of constructing a third mutually unbiased basis (MUB) in dimension d=6 (related to Zauner's conjecture). The proof introduces a novel algebraic framework based on the Schur-closure identity of the Gram matrix, yielding a fundamental factorized master equality. Through rigorous phase-alignment reduction and subresultant elimination, we demonstrate that the regular physical regime inevitably forces a "spectral collapse". This mechanism mathematically annihilates the Vandermonde determinant, strictly driving the physical phase vector to the trivial zero vector and yielding an absolute contradiction. Furthermore, we provide explicit Cramer coefficients, boundary-lifting lemmas, and a self-contained numerical witness certificate to ensure full mathematical traceability. Our results definitively show that the μ_5 branch carries no non-degenerate physical solution, establishing a major milestone toward the full algebraic resolution of the d=6 MUB problem.
This manuscript provides a complete algebraic proof excluding the μ_5 branch—the most challenging generic branch in the longstanding open problem of constructing a third mutually unbiased basis (MUB) in dimension d=6 (related to Zauner's conjecture). The proof introduces a novel algebraic framework based on the Schur-closure identity of the Gram matrix, yielding a fundamental factorized master equality. Through rigorous phase-alignment reduction and subresultant elimination, we demonstrate that the regular physical regime inevitably forces a "spectral collapse". This mechanism mathematically annihilates the Vandermonde determinant, strictly driving the physical phase vector to the trivial zero vector and yielding an absolute contradiction. Furthermore, we provide explicit Cramer coefficients, boundary-lifting lemmas, and a self-contained numerical witness certificate to ensure full mathematical traceability. Our results definitively show that the μ_5 branch carries no non-degenerate physical solution, establishing a major milestone toward the full algebraic resolution of the d=6 MUB problem.
Evidence for and against Zauner's MUB Conjecture in C6
The problem of finding provably maximal sets of mutually unbiased bases in Cd, for composite dimensions d which are not prime powers, remains completely open. In the first interesting case, d=6, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in d=6 by looking at their `shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers q≡5mod12, however, we are able to show some curious evidence which -- at least formally -- points in the opposite direction. In C6, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.
We present a detailed computational and algebraic study of Mutually Unbiased Bases (MUBs) in finite-dimensional Hilbert spaces, with a particular focus on dimensions 2, 3, 4, and the challenging case of 6. Starting from the Hadamard-phase parametrization, we derive explicit analytical conditions for mutual unbiasedness in dimension 4, providing a tractable system of trigonometric constraints on the phase parameters. We then explore a tensor-product construction via Pauli operators, highlighting the algebraic and group-theoretical origin of MUBs in two-qubit systems, and demonstrating how these constructions yield a complete set of 5 MUBs in dimension 4. Extending our approach, we investigate the Fourier-family method in dimension 6, where the absence of a prime-power structure imposes strong rigidity constraints and limits the known constructions to sets of 3 MUBs. We provide a systematic computational framework for testing candidate phase vectors, bridging the gap between analytical insight and numerical exploration. Finally, we generalize the discussion to arbitrary prime-power dimensions, emphasizing the role of finite-field structures, Heisenberg-Weyl operators, and discrete symmetries in generating complete sets of MUBs. Our work offers a transparent, line-by-line verification methodology, highlighting both the geometric and algebraic richness of MUBs, and clarifying why certain dimensions resist full analytical constructions. This study serves as a comprehensive resource for researchers seeking both theoretical understanding and practical construction of MUBs in quantum information science.
The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions d which are not prime powers, remains completely open. In the first interesting case,~d=6, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~d=6 by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~q≡5mod12, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.
Which pure states give uniform outcomes in several mutually unbiased measurements, and when can they form another measurement basis? We distinguish common unbiased states, their orthonormal completion, and further extension. For a displayed phase-deformed Fourier pair in dimension 2m, a clock symmetry and skew antiunitary complete every prescribed common unbiased state to an explicit orthonormal basis. In dimension six, an amplitude-profile obstruction proves strong unextendibility of a given mutually unbiased triple when its original third-basis rays are invariant under the ternary clock. We give an exact nonproduct example and an application to a companion whose columns are localized at single Fourier frequencies. For every exact Hadamard matrix in a specified closed entrywise neighborhood of a fixed reference matrix, the coordinate and Hadamard bases have exactly 48 common unbiased rays but no third measurement basis, with a quantitative lower bound on the squared Gram defect of every six-column candidate.
We also study auxiliary complex equations for common states, retaining algebraic multiplicities. For a flat unitary of order six with all 3×3 minors nonzero, the projective ray scheme is finite; each singular two-minor contributes two reduced isolated boundary points, and the affine coordinate algebra is finite-dimensional over C. In every order, a critical-algebra identity preserves multiplicities; in dimension six it gives finiteness for a fixed phase fibre. Specialization over a discrete valuation ring controls complete reciprocal-orthogonal collections. A nonphysical Tao/Potts collection separates algebraic from physical solutions. These results do not determine the global maximum number of mutually unbiased bases or classify all triples.
Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions 3, 4, and 5 demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension 6 where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.