We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F ( a , b ). However, in the main result of this paper we also prove that for any values of the parameters ( a , b ), the standard basis and F ( a , b ) cannot be extended to a MUB-quartet . The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.
We exhibit an infinite family of triplets of mutually unbiased bases (MUBs) in dimension 6. These triplets involve the Fourier family of Hadamard matrices, F ( a , b ). However, in the main result of this paper we also prove that for any values of the parameters ( a , b ), the standard basis and F ( a , b ) cannot be extended to a MUB-quartet . The main novelty lies in the method of proof which may successfully be applied in the future to prove that the maximal number of MUBs in dimension 6 is three.
We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.
We prove that the maximum number of mutually unbiased bases in complex dimension sixis three. The proof is structural and does not rely on exhaustive classification, numerical search,or probabilistic arguments. The key ingredients are: (1) a Laurent–polynomial obstructionblocking the Haagerup family of complex Hadamard matrices, (2) an analogous obstructionblocking the 2-circulant family, and (3) a structural Bridge Lemma showing that any complexHadamard matrix arising as a transition matrix in a MUB triplet must exhibit an opposite-pairphenomenon in its row-ratio multisets. Combined with the recognition theorem of Matszangoszand Sz¨oll˝osi, this forces all MUB-triplet transition matrices in dimension six into families alreadyobstructed, yielding a contradiction to the existence of a fourth mutually unbiased basis.
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1 bases in dimension d=q(q+1) for every prime power q≡3(mod4). This count grows as d and cannot be exceeded by tensor products of smaller sets.
In this research, I constructed an exact algebraic triple of mutually unbiased bases (MUBs) in ℂ⁶ consisting of the computational basis, a member D(ρ) of the Diţă slice from Karlsson's three-parameter family, and an explicit third basis B. Through exact symbolic computation over a degree-96 number field extension, I proved that both remaining transition matrices in the triple belong precisely to the two-parameter Fourier family F(a,b) up to equivalence and transposition, revealing that the Diţă-slice matrix factors directly into a product of two Fourier-type Hadamard matrices. Applying the theorem of Jaming et al., I established that both Fourier edges are non-extendable, proving the triple cannot be extended to any MUB quartet. Additionally, I derived a closed-form expression D(z) for the entire Diţă slice, resolved degeneracies in Karlsson's Möbius parametrization, and computed its 2×2 Hadamard submatrices exactly (51 generically and 75 at z=1), while identifying a second exact Fourier-Fourier triangle at z=1. Finally, using homotopy continuation to track witness systems across seven slice parameters, I recovered 40 third bases that all preserved this Fourier-Fourier pattern to residuals within 5×10⁻¹⁵, providing strong numerical backing for a global structural conjecture along the full Diţă slice.
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.
Abstract We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that each MUB-triplet in Cd C d is characterized by a d×d×d d × d × d object that we call a Hadamard cube . We describe the basic properties of Hadamard cubes and show how a MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.