Constructions of Mutually Unbiased Bases

Andreas Klappenecker (Texas A&M University); Martin Roetteler (University of Waterloo)

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Constructions of Mutually Unbiased Bases

Andreas Klappenecker (Texas A&M University), Martin Roetteler (University of Waterloo)

Quantum Physics · Submitted 15 Sep 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give a simplified proof of this fact based on the estimation of exponential sums.

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arXiv IDquant-ph/0309120 (version 1)

Pith ID53RJVT6X, Pith's permanent record of this paper and its review

CitationAndreas Klappenecker (Texas A&M University), Martin Roetteler (University of Waterloo). "Constructions of Mutually Unbiased Bases." arXiv:quant-ph/0309120 (2003). https://pith.science/paper/53RJVT6X

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.37 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.72 · compares with prior results 0.41 · mentions code or data availability 0.02.

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Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if |<b|b'>|^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if \|<b\|b'>\|^2=1/d holds for all b in B and b' in B'.	background 1.00	0.07	0.04	asserted 0.99
s1	The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1.	background 0.98	0.10	0.11	asserted 1.00
s2	If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist.	background 0.99	0.05	0.16	asserted 1.00
s3	We give a simplified proof of this fact based on the estimation of exponential sums.	contribution 0.63	0.94	0.17	asserted 1.00
s4	We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.	future work 0.55	0.63	0.05	unclear 0.67

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Original abstract

Two orthonormal bases B and B' of a d-dimensional complex inner-product space are called mutually unbiased if and only if |<b|b'>|^2=1/d holds for all b in B and b' in B'. The size of any set containing (pairwise) mutually unbiased bases of C^d cannot exceed d+1. If d is a power of a prime, then extremal sets containing d+1 mutually unbiased bases are known to exist. We give a simplified proof of this fact based on the estimation of exponential sums. We discuss conjectures and open problems concerning the maximal number of mutually unbiased bases for arbitrary dimensions.

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References

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Construction of Kochen-Specker Sets from Mutually Unbiased Bases
quant-ph 2025-09 conditional novelty 6.0 of 10

A systematic MUB-based enumeration yields a 69-ray 50-context KS nucleus unifying known constructions, plus forcing gadgets in D=4 and D=5 that enforce maximal unbiasedness.
Faithful real embedding of a three-dimensional complex Kochen-Specker configuration
quant-ph 2025-11 conditional novelty 4.0 of 10

Any finite set of complex 3D rays can be phase-adjusted to embed faithfully in real 6D, and the Cabello 165-ray KS set becomes colourable there.

How the literature uses it

12 arXiv papers cite this in 16 sentences

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Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Then a constructive proof of the existence of mutually unbiased bases for dimensions which are power of a prime is presented.

A new proof for the existence of mutually unbiased bases · 0.82

Papers citing it · 9 sentences without a confident label

Uses its method or tool 1 paper

As examples, we first consider two coupled qubits, and construct its temporal assemblage using five mutually unbiased bases (MUBs) [ref] .

Temporal Steering in Four Dimensions with applications to coupled qubits and magnetoreception · introduction · 1.00

Cites it as background 6 papers

Similarities between MUBs and mutually orthogonal Latin squares have been analyzed during the last years [ref] .

The correspondence between mutually unbiased bases and mutually orthogonal extraordinary supersquares · Introduction · 0.92
When $d$ is a prime power, i. e. $d = p^m$ for a prime $p$ and $m ∈$, it is known that $N(d) = d + 1$ and construction methods for complete sets of MUBs are known [ref] , while for non-prime-power dimensions $N(d)$ is unknown.

Complete sets of cyclic mutually unbiased bases in even prime power dimensions · Complete sets of MUBs · 0.86
The known methods for constructing standard MUBs span a wide range of mathematical frameworks including operator-algebraic approach based on the WH group [ref] ; combinatorial designs involving relative difference sets, finite semifields, and symplectic spreads [ref] ; finite geometry, mutually orthogonal Latin squares, affine and projective planes [ref] ; Galois rings [ref] .

Generalized Numerical Construction of MUBs: A Group Theoretical Investigation · Analytical Construction · 1.00

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