Why three? A two-level system with four mutually unbiased questions
Abstract
Two sharp yes/no questions are mutually unbiased if and only if their involutions anticommute. A set of unbiased questions is therefore a Clifford algebra, whose maximal size is odd, meaning four anticommuting questions typically imply a fifth: the two-level systems of real, complex and quaternionic quantum theory allow two, three and five questions, while four is skipped. Without this operator product restriction, a two-level system with four unbiased questions exists as the four-dimensional Bloch ball, the hyperbit. The single-system postulates that pick out the Jordan state spaces allow such a model, and only energy observability rules it out. We show what it lacks, and realise it inside two qubits: its Kirkwood-Dirac imaginary parts are hidden so(4) generators, its composites allow entanglement but no interaction, and its 24-cell of states stays Spekkens preparation contextual down to depolarising strength 2/3. In fermionic terms, the four questions are the Majorana operators of two modes while the fifth is parity: the four-question system is therefore the sector removed by parity superselection.
Related
- Sharp Total-Support Uncertainty for the Standard Complete Set of Mutually Unbiased Bases
- More mutually unbiased bases
- Using topology to reveal all mutually unbiased bases in dimension six
- Mutually unbiased bases: common states, completion, and extension obstructions
- Strict non-tightness of the level-3 quantum bootstrap for a natural three-dimensional Hamiltonian, at an excited level
BibTeX
@misc{hance2026three,
title = {{Why three? A two-level system with four mutually unbiased questions}},
author = {Jonte R. Hance},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10078},
eprint = {2609.10078},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10078},
}
RIS
TY - UNPB
TI - Why three? A two-level system with four mutually unbiased questions
AU - Jonte R. Hance
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10078
UR - https://arxiv.org/abs/2609.10078
AB - Two sharp yes/no questions are mutually unbiased if and only if their involutions anticommute. A set of unbiased questions is therefore a Clifford algebra, whose maximal size is odd, meaning four anticommuting questions typically imply a fifth: the two-level systems of real, complex and quaternionic quantum theory allow two, three and five questions, while four is skipped. Without this operator product restriction, a two-level system with four unbiased questions exists as the four-dimensional Bloch ball, the hyperbit. The single-system postulates that pick out the Jordan state spaces allow such a model, and only energy observability rules it out. We show what it lacks, and realise it inside two qubits: its Kirkwood-Dirac imaginary parts are hidden so(4) generators, its composites allow entanglement but no interaction, and its 24-cell of states stays Spekkens preparation contextual down to depolarising strength 2/3. In fermionic terms, the four questions are the Majorana operators of two modes while the fifth is parity: the four-question system is therefore the sector removed by parity superselection.
ER -