# Constructions of Mutually Unbiased Bases

@inproceedings{Klappenecker2003ConstructionsOM, title={Constructions of Mutually Unbiased Bases}, author={Andreas Klappenecker and Martin R{\"o}tteler}, booktitle={International Conference on Finite Fields and Applications}, year={2003} }

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…

## 272 Citations

ul 2 00 4 Mutually Unbiased Bases and Finite Projective Planes

- Mathematics
- 2017

It is conjectured that the question of the existence of a set of d + 1 mutually unbiased bases in a ddimensional Hilbert space if d differs from a power of prime is intimatelly linked with the…

Real Mutually Unbiased Bases

- Mathematics, Computer Science
- 2005

A simpler, alternative proof that there can be at most d/2+1 real mutually unbiased bases in dimension d instead of invoking the known results on extremal Euclidean line sets is given.

The limitations of nice mutually unbiased bases

- Mathematics, Computer Science
- 2004

It is shown that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches.

Mutually unbiased bases, spherical designs, and frames

- MathematicsSPIE Optics + Photonics
- 2005

The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased…

Weighted complex projective 2-designs from bases : Optimal state determination by orthogonal measurements

- Mathematics
- 2007

We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then…

Uniqueness of Sets of Mutually Unbiased Bases of Order 5

- Mathematics

It is known that a set of k mutually unbiased bases of order d is unique (to within equivalence) for d ∈ {2,3,4}; in particular this is true for complete sets of mutually unbiased bases (the case k =…

New construction of mutually unbiased bases in square dimensions

- Mathematics, Computer ScienceQuantum Inf. Comput.
- 2005

The construction combines the design-theoretic objects (s, k)-nets and generalized Hadamard matrices of size s to show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s2 provided that there are w mutually orthogonal Latin squares of order s.

There is no generalization of known formulas for mutually unbiased bases

- Mathematics
- 2003

In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕N. This…

On the Maximal Number of Real Mutually Unbiased Bases

- Mathematics
- 2005

In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real…

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There is no generalization of known formulas for mutually unbiased bases

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In a quantum system having a finite number N of orthogonal states, two orthonormal bases {ai} and {bj} are called mutually unbiased if all inner products ⟨ai∣bj⟩ have the same modulus 1∕N. This…

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