More mutually unbiased bases

arXiv:2609.40311 · quant-ph, math-ph, math.MP, math.OA · Submitted 2026-09-30 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "More mutually unbiased bases".

Mira: Mutually unbiased bases (MUBs) are crucial for quantum information tasks such as state reconstruction, entanglement detection, and quantum cryptography.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we've been looking at the paper "More mutually unbiased bases," and it seems like the main point is that they've developed an ansatz for constructing more mutually unbiased bases than what we thought was possible in many dimensions.

Mira: That makes sense; they introduce this specific method using a tensor product of Fourier matrices, possibly with a real Hadamard matrix, to build these bases.

Lev: From an error correction standpoint, if we could actually build these sets on hardware, the complexity of managing those different basis measurements would be something we'd have to seriously consider for fault tolerance.

Kai: Exactly what Mira said; it sounds like they are tackling a fundamental limitation in how many measurements we can perform reliably in quantum systems.

Mira: They claim this construction yields specific numbers, like five MUBs in dimension twelve six in dimensions forty-eight ninety-six and one hundred ninety-two and so on.

Lev: Those specific counts are interesting because they show concrete improvements over the standard tensor product bound T(d) for those particular dimensions.

Kai: That's what I want to explore further; if these results are solid, it suggests we might be able to design quantum circuits that probe a richer set of information than our current theoretical limits suggest.

Mira: The paper also points out that this construction allows the number of MUBs to exceed the tensor product bound asymptotically in every eighteenth dimension.

Lev: As an error correction researcher, I'd wonder how robust these constructions are when you start scaling up to larger dimensions where noise becomes a bigger factor.

Kai: That brings us to the more advanced part of their work; they also show that using Paley’s real Hadamard matrix can generate q plus one bases in dimension d equal to q times q plus one for every prime power q congruent to three modulo four.

Mira: That result is quite specific, showing a growth rate of the number of bases related to the square root of the dimension and that it can't be beaten by tensor products of smaller sets.

Lev: If we could harness Paley’s matrix effectively, that would open up a whole new class of constructions for generating bases in those specific dimensions.

Paper summary: Kai: So, they are showing multiple ways to get more MUBs, using different unitary matrices based on phased Fourier constructions or phased Fourier-Hadamard constructions.

Mira: The methodology relies on defining a unitary matrix U d based on a fixed unitary matrix multiplied by a diagonal phase matrix, and they explore choices like U d = Fd1 ⊗... ⊗ Fdk or involving H/sqrt(h).

Lev: I'm curious about the practical implications here; implementing these complex tensor products of Fourier matrices on actual physical qubits would be an engineering challenge that requires very precise control over phase shifts.

Kai: It sounds like the paper is laying out a framework that goes beyond just finding a few extra bases; they are building a general method for generating more MUBs in many dimensions.

Mira: The specific constructions they detail, like the five MUBs in dimension twelve using U12 = F6 ⊗ F2, show how this works in practice by defining specific phase vectors w(twelve).

Lev: I'd ask about the complexity of those phase vectors; ensuring you can actually prepare those specific state vectors on a physical system is a huge hurdle for experimental realization.

Kai: The paper also describes the construction for seven MUBs in dimension thirty-six using U36 = F cubed ⊗ F squared and defining specific indexing schemes to get those seven bases.

Mira: That indexing scheme, involving indices x, z, y1, and y2, shows how they systematically generate a set of vectors that yield the desired MUBs in dimension thirty-six.

Lev: When you're dealing with such structured sets of measurements on hardware, the fidelity requirements for each individual basis state would be incredibly stringent to maintain those counts.

Kai: Furthermore, they show that extending their constructions using complete prime-power sets can yield N(d) greater than six MUBs in dimensions like d = three times two to the power of n when n is four or more.

Mira: That extension shows how you can combine their initial findings with other known methods to push the limits even further beyond what they've established for those specific cases.

Lev: If we look at error correction, having a larger set of available measurements might give us more options for syndrome extraction, which could potentially help in correcting errors more effectively.

Kai: The paper also discusses how combining Paley’s matrix with a fixed real Hadamard matrix H0 can lead to an unbounded multiplicative advantage in the number of MUBs.

Paper summary: Mira: This part about combining Paley’s matrix Hq with a fixed H0 to create a larger Hadamard matrix shows a way to construct bases whose count grows based on the prime factorization of the dimension.

Lev: That kind of mathematical structure suggests that if we can find physical systems that naturally support these kinds of highly structured unitary operations, we might find new ways to encode quantum information robustly.

Kai: So, to wrap up for a second look at "More mutually unbiased bases," the authors present several ansatzes for constructing more MUBs using tensor products of Fourier matrices and potentially real Hadamard matrices.

Mira: The core claim is that this approach allows them to construct sets of MUBs that exceed the standard tensor product bound asymptotically in every eighteenth dimension, and they show specific counts like five in dimension twelve and ten in dimension six hundred forty-eight.

Lev: For the future, the real challenge lies in moving these highly structured mathematical constructions from theoretical constructs into physical systems where we can actually cool and measure these states to verify the results.

Kai: That's a good summary of what they achieved with "More mutually unbiased bases." It seems like a really deep dive into how structure in the unitary operations dictates the number of measurable quantum bases available.

Mira: The implications for quantum cryptography and state reconstruction are significant, because having access to more MUBs means potentially richer information processing capabilities in those areas.

Lev: I think if we can even get one of these specific constructions running on a small-scale system, it would provide valuable data on the limits of what's achievable before we try to scale up for real error correction applications.

Kai: It really seems like the work is pushing the boundaries of what we know about MUBs in higher dimensions by providing concrete, albeit complex, constructions.

Mira: The paper demonstrates that utilizing these specific tensor product structures gives a way to systematically generate more bases than previously known bounds suggested for certain dimensions.

Lev: We'll have to watch closely to see if these mathematical advantages translate into any practical benefits when we start discussing real hardware implementations and the necessary error mitigation strategies for such complicated measurements.

Conclusion: Kai: So, this paper, "More mutually unbiased bases," is all about finding ways to generate more mutually unbiased bases than we thought possible in many dimensions using tensor products of Fourier matrices and Hadamard matrices.

Mira: From a theoretical standpoint, what strikes me is how they construct these sets; it seems the methodology is deeply rooted in specific unitary transformations that leverage the structure of those matrix products to create these extra bases.

Lev: I'm thinking about the hardware side right away, because if we can actually build a system that implements these specific unitary matrices on physical qubits, managing all those different basis states would be quite a headache for error correction protocols.

Kai: That’s exactly what I want to know; what was actually built and measured to verify these claims? Did they show any experimental data?

Mira: The paper provides the mathematical framework and specific counts, which suggests a solid theoretical foundation for these constructions, even if the physical implementation details are still being explored.

Lev: For error correction researchers like us, the robustness of these sets is a huge question; we need to know how resilient they are when you start dealing with real-world noise and decoherence on actual hardware.

Kai: So, it seems they're pointing toward new ways to access richer quantum information through these more bases.

Mira: The implication is that our ability to reconstruct quantum states or detect entanglement might be enhanced if we can utilize these larger sets of MUBs in practical applications like cryptography.

Lev: If we can harness this mathematical structure, it could open up entirely new avenues for encoding information in a way that’s more resilient to certain types of errors.

Kai: That's a big deal, suggesting a potential pathway for more powerful quantum computation or communication protocols down the line.

Mira: We need to keep pushing on the underlying assumptions; what limitations does the paper themselves acknowledge about these constructions, especially when scaling up?

Lev: I'd like to hear if they flag any specific conditions under which these constructions fail, so we don't get excited about something that simply doesn't work in practice.

Kai: That’s fair; understanding the boundaries of this construction is just as important as knowing the potential advantages.

Mira: Exactly, so while the theory suggests an expansion beyond previous limits, we have to scrutinize the precise conditions under which those gains are realized.

Mateo C´ardenes Wuttig, Joseph Tindall

Department of Applied Physics, Yale University · Yale Quantum Institute · Center for Computational Quantum Physics, The Flatiron Institute

quant-ph, math-ph, math.MP, math.OA

Submitted: 2026-09-30

Updated: 2026-09-30

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 92/100

The gist: Mutually unbiased bases (MUBs) are crucial for quantum information tasks such as state reconstruction, entanglement detection, and quantum cryptography.

Key concepts

Mutually Unbiased Bases (MUBs)
MUBs are special sets of quantum measurements that are maximally non-commuting. They are crucial for quantum tasks like state reconstruction and cryptography. The paper focuses on finding ways to generate these sets in higher dimensions than standard methods allow.
Tensor Product Bound T(d)
This is the universal upper limit for the number of MUBs that can be obtained from a simple tensor product decomposition of dimension d. The new ansatz aims to create sets that exceed this bound, demonstrating a more efficient way to generate bases.
Paley’s Real Hadamard Matrix
This specific matrix is used in the construction to achieve an unbounded advantage. It allows for the creation of q + 1 bases in dimension d = q(q + 1) for prime powers q congruent to 3 (mod 4). This method shows a way to generate an increasing number of bases.
Phased Unitary Ansatz
The core construction involves a unitary matrix U_d defined by multiplying a fixed unitary matrix by a diagonal phase matrix. The ansatz explores different forms for U_d, such as those based on factorizations (Fourier) or combinations involving Hadamard matrices.

Terminology

Summary

Mutually unbiased bases (MUBs) are crucial for quantum information tasks such as state reconstruction, entanglement detection, and quantum cryptography. This work introduces an ansatz that constructs more MUBs than previously known in many dimensions by utilizing a tensor product of Fourier matrices and optionally a real Hadamard matrix. The findings demonstrate that this construction allows for the creation of sets of MUBs that exceed the standard tensor product bound asymptotically in every eighteenth dimension, and specifically shows a method using Paley’s real Hadamard matrix to construct an unbounded number of bases in dimensions related to prime powers.

Key Findings on MUB Counts

The paper presents specific constructions yielding more MUBs than the universal upper bound of 1 plus the minimum prime factor in the tensor product decomposition, denoted as T(d). The ansatz yields:

** 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.**

The paper also highlights a specific result concerning Paley’s real Hadamard matrix:

"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 (mod 4). This count grows as √d and cannot be exceeded by tensor products of smaller sets."

The Phased Unitary Ansatz

The core of the construction relies on a specific unitary matrix, denoted as U d, which is defined based on a fixed unitary matrix multiplied by a diagonal phase matrix. The ansatz considers bases of the form:

Aa = diag(wa)Ud,

where each vector wa has entries of magnitude one, and w1 = (1,..., 1) always gives A(d)1 = Ud.

The construction utilizes two primary choices for the unitary U d:

  1. Phased Fourier constructions for a factorization d = d1 · d2 ·…·dk, where U d = Fd1 ⊗ ··· ⊗ Fdk.

  2. Phased Fourier-Hadamard constructions, where U d = Fd1 ⊗ ··· ⊗ Fdk ⊗ H/√h, with H being a real Hadamard matrix of order h.

Specific Constructions and Extensions

The paper details several explicit constructions that exceed the tensor product bound T(d):

** Five MUBs in dimension 12,**

These are constructed using U12 = F6 ⊗ F2 and specific phase vectors w(12). The set is shown to give five MUBs, exceeding T(12) = 4.

The construction for seven MUBs in dimension 36 uses U36 = F cubed ⊗ F squared and specific phase vectors derived from a unique indexing scheme:

The set... gives seven mutually unbiased bases in d = 36, exceeding T(36) = 5.

For dimensions related to powers of two, six MUBs are obtained in dimensions 48, 96, and 192 using U d = F3 ⊗ F m2. The paper also shows how these constructions can be extended:

Tensoring our constructions here with complete prime-power sets... N(d) ≥ 6 = T(d) + 2, d = 3 · 2 n, n ≥ 4.

Unbounded Advantage from Hadamard Matrices

The Fourier-Hadamard construction is generalized for any real Hadamard matrix H of order h ≥ 4. The construction yields:

1 + λ(h − 1) many MUBs, where we define λ(n) = min p i for n = Q i p a i for the prime factorization n = Q i p a i of any integer n ≥ 2.

The paper proves an unbounded multiplicative advantage by combining Paley’s matrix with a fixed real Hadamard matrix H0:

**"Start with a real Hadamard matrix H0 of order h0, and fix a prime r > 2 that does not divide h0. For any L ≥ r, choose a prime q ≡ 3 (mod 4) such that q + 1 is divisible by every prime up to L, but not by r squared. Let Hq be Paley’s matrix of order q + 1 [27]. Then H0 ⊗ Hq is a real Hadamard matrix of order h = h0(q + 1). Every prime up to L divides h, so none divides h − 1.

Improvements for AI systems

Here are the specific improvements to AI systems that can be derived from this scientific paper, categorized by capability:

)1. Enhanced Quantum State Tomography and Verification:

The paper explicitly details constructions for Mutually Unbiased Bases (MUBs) in high dimensions (e.g., dimension 648, 756). In the context of quantum information science (Section EXPERIMENTAL IMPLICATIONS), these MUBs are used for:

  • Detecting entanglement and certifying dimension.

  • Verifying maximally entangled states efficiently.

The improved AI system could perform:

  • Perform near-optimal quantum state tomography on high-dimensional qubits or qudits, using the constructed MUB sets as measurement bases. This allows for more robust certification of the system's state fidelity than standard tomography methods, especially in dimensions where the tensor product bound is exceeded.

  • Develop algorithms to certify high-dimensional entanglement using only a small set of MUB measurements (as demonstrated by recent experimental work cited in [33]), making entanglement certification faster and less resource-intensive for large systems.

)2. Optimized Quantum Cryptography Protocols:

MUBs are fundamental to quantum cryptography protocols (mentioned in the Introduction). The ability to construct and utilize sets of MUBs that exceed the tensor product bound provides a significant advantage in security analysis:

  • Design quantum key distribution (QKD) or secure communication protocols that leverage these larger MUB sets. This would allow for testing security against eavesdropping across a wider set of measurement bases simultaneously, increasing the robustness of the security proof against unknown basis choices.

  • Implement higher-dimensional quantum cryptography schemes where the number of available non-commuting measurement operators is maximized, leading to potentially stronger information theoretic bounds on eavesdropping detection.

)3. High-Dimensional Quantum Simulation and Machine Learning:

The paper's focus on complex unitary matrices (tensor products of Fourier matrices and Hadamard matrices) suggests a strong foundation for simulating quantum systems:

  • Develop quantum simulators capable of representing and manipulating high-dimensional Hilbert spaces efficiently by leveraging the structure of the MUB bases. The specific ansatz used (fixed unitary multiplied by a diagonal phase matrix) provides a structured way to explore the state space.

  • Train machine learning models (e.g., variational quantum circuits) on data encoded in these high-dimensional quantum states, using the constructed MUBs for systematic sampling and measurement, potentially leading to more accurate and robust representations of complex quantum phenomena.

)4. Novel Quantum Algorithm Design:

The construction methods—particularly the use of Paley's matrix in specific prime power dimensions—suggest new algorithmic approaches:

  • Design quantum algorithms that exploit the structure of these non-tensor product MUB sets for faster computation, such as Grover's search or Shor's algorithm variants tailored for higher-dimensional systems.

  • Develop new methods for quantum phase estimation or state preparation where the search space is structured according to the MUB geometry, potentially offering speedups over traditional approaches in specific dimensions like those derived from Paley constructions.

)5. Robust Theoretical Frameworks for Quantum Information:

The mathematical results concerning the asymptotic gain (Section III) provide a new theoretical benchmark:

  • Establish rigorous upper and lower bounds on the maximum number of MUBs in arbitrary non-prime-power dimensions, moving beyond the known tensor product limits. This provides a more precise theoretical landscape for quantum information theory.

  • Formalize asymptotic gains in quantum complexity classes, providing new tools to analyze how computational power scales when measurement bases are chosen optimally across exponentially large Hilbert spaces.

Abstract

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 4. This count grows as sqrt d and cannot be exceeded by tensor products of smaller sets.

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