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Mutually Unbiased Bases in Dimension Twelve: Tensor Saturation, Strong Unextendibility, and the Flat-Sum Trichotomy

Han-Jun Lim

Published Jun 13, 2026• 0 citations
Publisher

Abstract

Abstract Setting. The maximal number N(12) of mutually unbiased bases (MUBs) of C12 is unknown; no composite non-prime-power dimension has a known exact value. The quantum MacNeish (tensor) bound gives N(12) ≥ min{N(3),N(4)} = 4, and within Array Cosmology (AC) the integer 4 = k is the ququart rank of the rendering algebra R12 = M3(C)⊗ M4(C). This paper maps the d = 12 MUB landscape and states the tensor saturation conjecture N(12) = 4. Canonical sector. (i) Strong unextendibility [A]: no vector of C12 is unbiased to all four bases of the canonical product quadruple; the proof closes through an entanglement-forcing theorem, a flatness–circle-structure reduction, a monomial-rigidity step, and an atomic lemma whose ninebranches collapse to f = U+UV+V withexactminimum5/4 = (k+1)/k > 1. (ii)Localrigidity[A]: the canonical quadruple is rigid modulo its 191-dimensional gauge orbit; the Jacobian has exact rank 385 and exact nullity 191, certified by an explicit integer basis of the kernel verified in exact arithmetic, by modular arithmetic at p = 1000033, and by an independent gauge-dimension proof

DOI
10.5281/zenodo.20675928

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AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.

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The Maximum Number of Mutually Unbiased Bases in Dimension Ten is Three - PDF

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More mutually unbiased bases

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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 $q\equiv3\pmod4$. This count grows as $\sqrt d$ and cannot be exceeded by tensor products of smaller sets.

quant-pharXiv:2609.40311•Oct 1, 2026•0 citations
Abstract Setting. The maximal number N(12) of mutually unbiased bases (MUBs) of C12 is unknown; no composite non-prime-power dimension has a known exact value. The quantum MacNeish (tensor) bound gives N(12) ≥ min{N(3),N(4)} = 4, and within Array Cosmology (AC) the integer 4 = k is the ququart rank of the rendering algebra R12 = M3(C)⊗ M4(C). This paper maps the d = 12 MUB landscape and states the tensor saturation conjecture N(12) = 4. Canonical sector. (i) Strong unextendibility [A]: no vector of C12 is unbiased to all four bases of the canonical product quadruple; the proof closes through an entanglement-forcing theorem, a flatness–circle-structure reduction, a monomial-rigidity step, and an atomic lemma whose ninebranches collapse to f = U+UV+V withexactminimum5/4 = (k+1)/k > 1. (ii)Localrigidity[A]: the canonical quadruple is rigid modulo its 191-dimensional gauge orbit; the Jacobian has exact rank 385 and exact nullity 191, certified by an explicit integer basis of the kernel verified in exact arithmetic, by modular arithmetic at p = 1000033, and by an independent gauge-dimension proof
AbstractWe prove that the maximum number of mutually unbiased bases in C6 is exactly three:M(6) = 3. The proof exploits the tensor product structure C6 ∼= C2 ⊗ C3 and proceedsby showing that any vector unbiased to three canonical product bases must itself be aproduct vector. The key technical ingredient is a rigidity theorem: vectors in C3 that aresimultaneously flat in two specific bases form a finite six-element orbit. Combined withcoupling constraints from the tensor structure, this forces separability, contradicting therequirement that a fourth basis contain no product vectors. The argument is elementary,using only linear algebra and explicit polynomial system analysis.The author blasts Hungarian music from the 17th centurywhile memeing on visions of heretics in Japan btw.
AbstractWe prove that the maximum number of mutually unbiased bases in dimension d = 2p, wherep ≥ 3 is prime, equals exactly three: M(2p) = 3. The proof combines tensor product saturationwith a new rigidity theorem: any unimodular sequence on Zp that is flat in both the Fourier basisFp and the Weyl-Heisenberg basis Cp must be a quadratic chirp. This “double flatness” rigidity,proved via character-theoretic arguments on autocorrelation functions, forces all triply-unbiasedstates to be product states, contradicting the existence of a fourth MUB. No restriction to rootsof unity is required. This resolves the MUB problem for infinitely many composite dimensions
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.
On first sight, the number of mutually unbiased bases (MUBs) in d dimensions and the number of mutually orthogonal Latin squares (MOLS) of order d seem to be closely related. A careful look however reveals the connections cannot be really strong. In this talk a few similarities and differences between MUBs and MOLS are presented. The first similarity involves their maximal numbers. There are at most d + 1 MUBs and also d + 1 squares in an augmented set of MOLS. This number is attained if d is a power of a prime and we give an algorithm which relates MUBs and MOLS in these dimensions [1]. Another similarity follows from considerations of composite d. One can think about a Hilbert space of a composite dimension as describing several subsystems of prime-power dimensions. Since we can always build MUBs in the global space as tensor products of MUBs for the subspaces, there are at least M + 1 of them for the whole system, where M is the smallest prime-power factor of d. A similar construction exists for MOLS: squares of composite orders can be built as direct products of squares of prime-power orders. The corresponding lower bound on the number of MOLS is called the MacNeish’s bound. In general, both the MacNeish’s bound and the number of MUBs which follows from the tensor product construction are known not to be tight. However, the simplest example today involves dimension d = 676 for MUBs [2], whereas MacNeish’s bound is not tight already for d = 10. We therefore checked whether the connection [1] also works for d = 10, and found it no longer links all MOLS to MUBs [3]. The general results of Ref. [4] reveal in this context that the connection fails whenever MacNeish’s bound is not tight. The last (apparent) similarity to be presented is related to incomplete sets of MOLS and MUBs. There is a Latin square of order 4 which does not have an orthogonal mate (so called bachelor Latin square) and connection [1] links it with a set of MUBs which also cannot be extended. It might be tempting to conjecture that every unextendible augmented set of MOLS has a corresponding unextendible set of MUBs. We already disproved such conjecture for the connection [1] and now we briefly describe that there is no other connection for which the conjecture would hold true. One can see this in at least two ways [5]. The first involves bachelor Latin square of order 5. It was recently shown that bachelor Latin squares exist for all orders d > 3 [6]. If there is a strong relation between MOLS and MUBs, there should correspondingly exist a set of 3 MUBs in all dimensions d > 3. To the contrary, it is known that for d = 5 every set of 3 MUBs can be extended to a complete set of 6 MUBs [7]. The second way is more dramatic. It turns out that for d = 6 there exists a set of two MUBs which cannot be extended any further (and six is the smallest dimension in which this happens). Since an augmented set of MOLS contains at least 3 elements, the strong relation is excluded. For more arguments against a close connection between MOLS and MUBs see Ref. [8].
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+1q+1q+1 bases in dimension d=q(q+1)d = q(q+1)d=q(q+1) for every prime power q≡3(mod4)q\equiv3\pmod4q≡3(mod4). This count grows as d\sqrt dd​ and cannot be exceeded by tensor products of smaller sets.