More mutually unbiased bases

AI summary

Overview

Research area: Quantum information theory — specifically mutually unbiased bases (MUBs), complex Hadamard matrices, and high-dimensional quantum measurement.

Technical level: Advanced. The paper assumes familiarity with unitary matrices, tensor products, Fourier matrices, Hadamard matrices, and the standard MUB upper bound.

Scope: The paper introduces a simple "phased unitary" ansatz that produces explicit MUB sets exceeding the tensor product bound in many composite dimensions, including a general construction from any real Hadamard matrix.

What This Paper Is About

A set of MUBs describes quantum measurements that are as incompatible as possible: certainty in one basis gives uniformly random outcomes in any other. The maximum number of such bases, N(d), is known exactly only when the dimension d is a power of a prime, where it equals d+1; for other dimensions even the existence of four MUBs in dimension six is unresolved. This paper asks whether more MUBs can be constructed in non-prime-power dimensions than the standard tensor product bound T(d) = 1 + min_i p_i^{a_i} (for d = ∏ p_i^{a_i}) allows, and answers yes in many cases.

Key Contributions

1. A phased unitary ansatz. Every basis is written as A_a = diag(w_a) U_d, where U_d is a fixed unitary with entries of magnitude 1/√d and w_a is a vector of unit-modulus phases. The MUB condition reduces to a single equation on phase combinations, Eq. (4), turning basis construction into a phase-finding problem.
2. New explicit MUB counts in specific dimensions. The authors report five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648.
3. A universal Hadamard-to-MUB construction. Any real Hadamard matrix of order h ≥ 4 yields 1 + λ(h−1) MUBs in dimension d = h(h−1), where λ(n) is the smallest prime-power factor of n.
4. A Paley family with √d scaling. For every prime power q ≡ 3 (mod 4), Paley's Hadamard matrix gives q+1 MUBs in dimension d = q(q+1), a count that grows as √d and provably cannot be matched by tensor products of smaller MUB sets.

Main Findings

• Five MUBs in dimension 12. Using U_12 = F_6 ⊗ F_2 and ζ = e^{iπ/3}, the three phase vectors in Eqs. (8)–(10) together with the identity give five MUBs, exceeding T(12) = 4. This is the first improvement reported below dimension 100 relative to the tensor product bound, which previously gave the largest known sets in non-prime-power dimensions below 100.
• Seven MUBs in dimension 36. With U_36 = F_3^{⊗2} ⊗ F_2^{⊗2} and phases built from u and v in Eqs. (11)–(12), the set {I_36, A_1, …, A_6} gives seven MUBs, exceeding T(36) = 5.
• Six MUBs in dimensions 48, 96, and 192. For d = 3·2^m with m = 4, 5, 6 and U_d = F_3 ⊗ F_2^{⊗m}, six MUBs are obtained, against T(d) = 4.
• Seven MUBs in dimension 108. Using U_108 = F_3^{⊗3} ⊗ F_2^{⊗2}, seven MUBs are obtained instead of five.
• Ten MUBs in dimension 648. With d = h²/2, h = 4·3² = 36, a Paley Hadamard matrix of order 36, U_648 = (H/6) ⊗ F_2 ⊗ F̄_3^{⊗2}, and phase vectors derived from the Abel–Colbourn–Wojtas difference matrix, ten MUBs are obtained against T(648) = 9.
• General Hadamard construction. Taking U_d = F̃_{h−1} ⊗ H/√h in dimension d = h(h−1) yields 1 + λ(h−1) MUBs. This exceeds T(h(h−1)) = 1 + min{λ(h), λ(h−1)} exactly when λ(h−1) > λ(h). Orders h = 1 and h = 2 are handled by first enlarging H to a Hadamard matrix of order 4h.
• Paley scaling result. For prime powers q ≡ 3 (mod 4), dimension d = q(q+1) admits q+1 exactly unbiased bases, growing as √d. The example q = 27 gives 28 MUBs in dimension 756, compared with T(756) = 5. The authors note that Kumar and Maitra previously constructed q+1 approximately unbiased real bases in the same dimensions; these bases are exactly unbiased.
• Use of the order-668 Hadamard matrix. Applying the construction to the recently announced Hadamard matrix of order 668 by Alpöge and collaborators gives N(d) ≥ 24 > T(d) = 5 at d = 445,556 = 668·667.
• Tensor extensions. Tensoring with complete prime-power sets gives N(d) ≥ 6 = T(d)+2 for d = 3·2^n with n ≥ 4; N(d) ≥ 7 = T(d)+2 for d = 4·3^n with n ≥ 2; and N(d) ≥ 10 = T(d)+1 for d = 8·3^n with n = 4 or n ≥ 6. These exceed T(d) throughout d = 24n+12, d = 144n+48, and d = 144n+96, hence in at least one dimension in eighteen asymptotically.
• Tensor products cannot match the Paley family. Any factorization of d = q(q+1) into at least two factors greater than one contains a factor of dimension at most q, so ordinary tensor products cannot exceed q+1 bases, even if complete sets existed in the factors.
• Unbounded relative advantage. By tensoring a fixed real Hadamard matrix with suitable Paley matrices, one obtains orders h where λ(h−1) grows without bound while λ(h) stays bounded, so the ratio (1 + λ(h−1))/T(h(h−1)) grows without bound. The Supplemental Material proves this via Dirichlet's theorem and the Chinese remainder theorem.
• A continuous family in dimension 12. The five MUBs in d = 12 belong to a continuous family parameterized by θ ∈ ℝ, generated by a unitary of the form I_12 + (e^{iθ}−1)vv† with v = (e_2 − e_7)/√2.
• Not reported. The paper does not report an experimental demonstration, numerical search runtimes, or a general algorithm for finding the phase vectors; explicit phases are provided for the listed dimensions, with additional vectors in the Supplemental Material.

Methodology in Plain English

The starting point is a bookkeeping trick. If each basis is a fixed unitary matrix with a different diagonal matrix of phases attached to it, then checking whether two bases are mutually unbiased reduces to checking that a certain matrix built from the product of their phases has all entries of magnitude 1/√d. That is a finite set of algebraic conditions on phases rather than a search over arbitrary bases.

The authors choose the fixed unitary to be a tensor product of Fourier matrices, which handles prime-power factors, and optionally a real Hadamard matrix, which handles a factor of order h. Because Fourier and Hadamard matrices already have flat entries, the remaining work is to pick phases that satisfy the conditions. For each reported dimension they write the phases down explicitly, as closed-form expressions in the binary, ternary, or mixed-radix index of the row, and verify the MUB condition by direct substitution.

For the general Hadamard construction, the phases take a quadratic form resembling a finite-field Gauss sum: an exponential of (a−1)(xy − (h/4)y²)/s for the Fourier rows and (a−1)x²/s for the Hadamard row. For composite h−1, one finite field is used per prime-power factor, and prime-power factors are handled with small integer matrices. The acknowledgements state that the LLM systems ChatGPT 6 Astra and Claude Fable 5.1 were prompted to construct previously unknown MUBs, that the authors simplified and summarized the results, that they pushed the LLMs to identify a universal construction from any real Hadamard matrix, and that they independently checked all LLM-assisted material and take full responsibility for the content.

Why This Matters

Exact values of N(d) are unknown for every non-prime-power dimension, so each new explicit construction tightens the lower bound and constrains conjectures about the true maximum. Improvements at dimensions 12, 36, 48, 96, 108, 192, and 648 shift the frontier of a problem that has resisted resolution since the question of four MUBs in dimension six was raised. The √d scaling of the Paley family is qualitatively stronger than the fixed advantage offered by tensor products, and the proof that no tensor product of smaller sets can match it changes what is expected of tensor-based constructions generally.

Applications:

• Quantum state reconstruction. Each additional basis supplies d−1 independent probabilities, so more MUBs mean fewer measurement settings for full tomography.
• Entanglement detection and dimension certification. MUB measurements underpin tests that certify high-dimensional entanglement with few settings, already demonstrated with photons encoded in space.
• Quantum cryptography. MUB complementarity is the basis of prepare-and-measure and entanglement-based protocols.
• Verification of maximally entangled states. MUBs allow efficient verification protocols.

Industry relevance: the paper points out that programmable silicon photonic chips can already perform arbitrary local measurements in dimension twelve, so the five MUBs found here can be tested on existing hardware. More broadly, the results matter to developers of high-dimensional photonic quantum communication, quantum key distribution, and device benchmarking, where the number of required measurement settings directly affects cost and speed.

Future Directions

• Dimension six and beyond. The paper lists dimension six as the outstanding open case and names ten and twelve as the next highest unresolved dimensions. Whether the ansatz can produce four MUBs in dimension six is not resolved here.
• Experimental test in dimension 12. The five MUBs in d = 12 are proposed as a candidate for programmable silicon photonic hardware to certify entanglement more robustly.
• Exploiting the Hadamard conjecture. If the Hadamard conjecture holds, the construction would give MUBs in d = h(h−1) for every order h divisible by four, which the paper flags as a route to further results.
• Structural questions. The continuous family in dimension 12 raises the question of how these sets relate to known equivalence classes of MUBs and complex Hadamard matrices, and whether the ansatz can be generalized beyond Fourier and Hadamard factors.

Target Audience

Quantum information theorists working on MUBs, complex Hadamard matrices, and finite-field constructions; researchers in high-dimensional quantum communication and photonic quantum information who need concrete measurement sets; and graduate students with a background in linear algebra and quantum measurement who want a worked example of how algebraic ansätze produce new combinatorial constructions.

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