Using topology to reveal all mutually unbiased bases in dimension six
Abstract
Mutually unbiased bases (MUBs) describe complementary quantum measurements: certainty in one basis gives no information about the others. In dimension six, triples of MUBs are known, but whether a fourth basis could exist remained unresolved for over three decades. OpenAI recently reported a proof that none exists. Here we use the topology of the space of complex Hadamard matrices, which describe transitions between MUBs, to characterize all sets in dimension six. We prove that every mutually unbiased triple belongs to a two-parameter family or is a single isolated triple built from the Fourier matrix. Since none of them extend to a fourth basis, this also proves that no four MUBs exist. Our proof rests on a new topological result that all order-six complex Hadamard matrices except Tao's form a connected space. Connectedness reduces the classification of triples to finitely many overlapping four-dimensional phase balls. We certify with interval arithmetic that these balls cover the space and for every pair of MUBs represented in a ball, we then determine all third bases that extend it. Dimension six is now the first non-prime-power dimension in which all sets of MUBs are classified.
Related
- Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases
- More mutually unbiased bases
- Mutually unbiased bases: common states, completion, and extension obstructions
- Why three? A two-level system with four mutually unbiased questions
- Coanalytic subfields and bases of prescribed dimension
BibTeX
@misc{wuttig2026using,
title = {{Using topology to reveal all mutually unbiased bases in dimension six}},
author = {Mateo Cárdenes Wuttig and Joseph Tindall},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06917},
eprint = {2611.06917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06917},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Using topology to reveal all mutually unbiased bases in dimension six
AU - Mateo Cárdenes Wuttig
AU - Joseph Tindall
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06917
UR - https://arxiv.org/abs/2611.06917
AB - Mutually unbiased bases (MUBs) describe complementary quantum measurements: certainty in one basis gives no information about the others. In dimension six, triples of MUBs are known, but whether a fourth basis could exist remained unresolved for over three decades. OpenAI recently reported a proof that none exists. Here we use the topology of the space of complex Hadamard matrices, which describe transitions between MUBs, to characterize all sets in dimension six. We prove that every mutually unbiased triple belongs to a two-parameter family or is a single isolated triple built from the Fourier matrix. Since none of them extend to a fourth basis, this also proves that no four MUBs exist. Our proof rests on a new topological result that all order-six complex Hadamard matrices except Tao's form a connected space. Connectedness reduces the classification of triples to finitely many overlapping four-dimensional phase balls. We certify with interval arithmetic that these balls cover the space and for every pair of MUBs represented in a ball, we then determine all third bases that extend it. Dimension six is now the first non-prime-power dimension in which all sets of MUBs are classified.
ER -