Constructions of Mutually Unbiased Bases

By Andreas Klappenecker, Martin Rötteler. International Conference on Finite Fields and Applications, 2003

310 citations recorded by the source corpus.

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 ∈ B and b′ ∈ 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.

Subfields: Algebra and Number Theory, Artificial Intelligence, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/b1eeaf49c8e34a5f40c66802f0b1cea95befaf79

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Ross Duncan
2. Ingemar Bengtsson
3. Karol Zyczkowski
4. Wojciech Tadej
5. Bob Coecke
6. Shayne Waldron

In Artificial Intelligence

1. Bob Coecke
2. Ross Duncan
3. Ingemar Bengtsson
4. Karol Zyczkowski
5. Wojciech Tadej
6. Shayne Waldron
7. Miriam Backens
8. Edward Grefenstette

In Discrete Mathematics and Combinatorics

1. Ross Duncan
2. Ingemar Bengtsson
3. Karol Zyczkowski
4. Wojciech Tadej
5. Bob Coecke
6. Shayne Waldron

Citations in the corpus

• Interacting Quantum Observables (2008, methodology): “[10]); (iii) we constructed closed CCSs on C in FdHilbwp for all n.”
• On the construction of equiangular frames from graphs (2009, background): “When such frames do not exist (cf [STDH07]) closely related classes of frames have been suggested as substitutes, most notably Grassmannian frames and mutually unbiased bases (see [SH03], [BK06], [ KR04 ]).”
• A Concise Guide to Complex Hadamard Matrices (2005, background): “(2) If the dimension N is prime or a power of prime the number of maximally unbiased bases (MUB) is equal to N + 1 [51, 52], but for other dimensions the answer to this question is still unknown [53, 54, 55].”
• Geometry of Quantum States: Index (2007, methodology): “Later references have made the construction more transparent and more general [136, 535].”