Mutually unbiased bases and Hadamard matrices of order six
Open accessAbstract
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.
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In the N = {2,3,5} case, all CHMs are equivalent to the respective FT matrix, while for N = 4 there exists a one parameter equivalence class.
“In the N = { 2 , 3 , 5 } case, all CHMs are equivalent to the respective FT matrix, while for N = 4 there exists a one parameter equivalence class.”
From Universal linear optics (2015) · cites this paper as [56]
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Ozols, M. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1508.00860 · Cited by 3 · see citing works
- Authors
- Bengtsson, I., Bruzda, W., Ericsson, Å., Larsson, J.-Å., Tadej, W., & Życzkowski, K.
- Affiliations
- Stockholm University (SE)
- Jagiellonian University (PL)
- Linköping University (SE)
- Cardinal Stefan Wyszynski University in Warsaw (PL)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 2007-05-01
- Volume
- 48
- Issue
- 5
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W1983735761
- Cited by
- 130
- References
- 24
- Citations by year
- 2026: 4 · 2025: 3 · 2024: 3 · 2023: 2 · 2022: 6 · 2021: 5 · 2020: 4 · 2019: 2 · 2018: 5 · 2017: 4 · 2016: 8 · 2015: 9 · 2014: 9 · 2013: 10 · 2012: 13
- FWCI
- 16.16
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Jagiellonian University Repository (Jagiellonian University) (submittedVersion, other-oa)
- DiVA (Linkoping University) (submittedVersion, other-oa)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Coding theory and cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- mutually unbiased bases, complex Hadamard matrices, discrete Fourier transform
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