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Mutually Unbiased Bases in Composite Dimensions – A Review

Quantum Science and Technology (arXiv overlay)

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⚡ Quantum Brief

A 40-year-old open problem in quantum theory remains unsolved: whether complete sets of mutually unbiased bases (MUBs) exist in composite Hilbert space dimensions like six or ten, unlike in prime-power dimensions where solutions are known. Researchers McNulty and Weigert present 14 mathematically equivalent formulations of the existence problem, consolidating analytic, numerical, and computer-aided approaches to tackle this fundamental gap in quantum information theory. The review highlights stagnant progress despite growing applications of MUBs in quantum benchmarking, state tomography, and cryptography, as evidenced by rising arXiv preprints mentioning MUBs in abstracts but not titles. In dimension six—the smallest unsolved case—only three MUBs have been found, while a maximal set of seven remains unproven, underscoring the challenge of extending results beyond prime-power dimensions. The paper outlines potential solution strategies, linking MUBs to Hadamard matrices, finite geometries, and quantum designs, while emphasizing the problem’s deep connections to Bohr’s complementarity principle.

Mutually Unbiased Bases in Composite Dimensions – A Review

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AbstractMaximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.Featured image: Number of preprints uploaded to arXiv.org annually over the last 25 years in the sections computer sciences, mathematics and physics containing the expression ‘mutually unbiased’ in the title (dark) and in the abstract (light), respectively. The increasing frequency of the expression in the abstracts seems to indicate widespread use of mutually unbiased bases in applications such as benchmarking; the lack of fundamental progress related to the existence problem is consistent with fewer papers containing the expression ‘mutually unbiased’ in the title.Popular summaryThis review concerns a mathematical problem that has remained open for almost forty years. It is particularly important in the context of quantum theory and quantum information, as demonstrated early on by the influential work of Wootters and Fields on optimal quantum state determination. The main question is easy to state: how many orthonormal bases can exist in a finite-dimensional (typically complex) Hilbert space such that every pair is mutually unbiased? Mutual unbiasedness means that all basis vectors appear with equal weight when any vector from one such basis is expanded in another mutually unbiased basis. This highly symmetric constraint can be thought of as a quantitative description of Bohr’s principle of complementarity. It is known that only a limited number of mutually unbiased bases can exist in any given dimension. The allowed maximum number of bases can be constructed whenever the dimension of the Hilbert space is a prime or a power of a prime. In all other cases — namely composite dimensions — the existence of maximal sets remains open. In dimension six, the smallest open case, no more than three mutually unbiased bases have been found so far, and a proof that a maximal set of seven bases does — or does not — exist, remains elusive. Being connected to several other topics in mathematics and quantum theory, the existence problem can be reformulated in many different ways. This review aims to consolidate what is known about mutually unbiased bases in composite dimensions, by surveying the analytic, numerical, and computer-assisted approaches that have been developed to study it.► BibTeX data@article{McNulty2026mutuallyunbiased, doi = {10.22331/q-2026-04-01-2051}, url = {https://doi.org/10.22331/q-2026-04-01-2051}, title = {Mutually {U}nbiased {B}ases in {C}omposite {D}imensions – {A} {R}eview}, author = {McNulty, Daniel and Weigert, Stefan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2051}, month = apr, year = {2026} }► References [1] S. Aaronson. ``The NEW ten most annoying questions in quantum computing'' (May 13th, 2014). Shtetl-Optimized blog post. https://www.scottaaronson.com/blog/?p=1792 [2] K. Abdukhalikov, E. Bannai, and S. 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Princeton University Press. (1955). https://doi.org/10.1515/9781400889921Cited byCould not fetch Crossref cited-by data during last attempt 2026-04-01 12:32:48: Could not fetch cited-by data for 10.22331/q-2026-04-01-2051 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-04-01 12:32:49: No response from ADS or unable to decode the received json data when getting the list of citing works.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractMaximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.Featured image: Number of preprints uploaded to arXiv.org annually over the last 25 years in the sections computer sciences, mathematics and physics containing the expression ‘mutually unbiased’ in the title (dark) and in the abstract (light), respectively. The increasing frequency of the expression in the abstracts seems to indicate widespread use of mutually unbiased bases in applications such as benchmarking; the lack of fundamental progress related to the existence problem is consistent with fewer papers containing the expression ‘mutually unbiased’ in the title.Popular summaryThis review concerns a mathematical problem that has remained open for almost forty years. It is particularly important in the context of quantum theory and quantum information, as demonstrated early on by the influential work of Wootters and Fields on optimal quantum state determination. The main question is easy to state: how many orthonormal bases can exist in a finite-dimensional (typically complex) Hilbert space such that every pair is mutually unbiased? Mutual unbiasedness means that all basis vectors appear with equal weight when any vector from one such basis is expanded in another mutually unbiased basis. This highly symmetric constraint can be thought of as a quantitative description of Bohr’s principle of complementarity. It is known that only a limited number of mutually unbiased bases can exist in any given dimension. The allowed maximum number of bases can be constructed whenever the dimension of the Hilbert space is a prime or a power of a prime. In all other cases — namely composite dimensions — the existence of maximal sets remains open. In dimension six, the smallest open case, no more than three mutually unbiased bases have been found so far, and a proof that a maximal set of seven bases does — or does not — exist, remains elusive. Being connected to several other topics in mathematics and quantum theory, the existence problem can be reformulated in many different ways. This review aims to consolidate what is known about mutually unbiased bases in composite dimensions, by surveying the analytic, numerical, and computer-assisted approaches that have been developed to study it.► BibTeX data@article{McNulty2026mutuallyunbiased, doi = {10.22331/q-2026-04-01-2051}, url = {https://doi.org/10.22331/q-2026-04-01-2051}, title = {Mutually {U}nbiased {B}ases in {C}omposite {D}imensions – {A} {R}eview}, author = {McNulty, Daniel and Weigert, Stefan}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2051}, month = apr, year = {2026} }► References [1] S. Aaronson. ``The NEW ten most annoying questions in quantum computing'' (May 13th, 2014). Shtetl-Optimized blog post. https://www.scottaaronson.com/blog/?p=1792 [2] K. Abdukhalikov, E. Bannai, and S. 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Mutually Unbiased Bases in Composite Dimensions – A Review

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