# Mutually Unbiased Bases and the Complementarity Polytope

@article{Bengtsson2005MutuallyUB, title={Mutually Unbiased Bases and the Complementarity Polytope}, author={Ingemar Bengtsson and {\AA}sa Ericsson}, journal={Open Systems \& Information Dynamics}, year={2005}, volume={12}, pages={107-120} }

A complete set of N + 1 mutually unbiased bases (MUBs) forms a convex polytope in the N2 − 1 dimensional space of N × N Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N = pk, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N2 facets…

## 52 Citations

A gap for the maximum number of mutually unbiased bases

- Physics, Mathematics
- 2009

A collection of (pairwise) mutually unbiased bases (in short: MUB) in d > 1 dimensions may consist of at most d + 1 bases. Such “complete” collections are known to exists in C when d is a power of a…

ON MUTUALLY UNBIASED BASES

- Mathematics, Physics
- 2010

Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased…

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

- Mathematics, Physics
- 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…

Hjelmslev geometry of mutually unbiased bases

- Mathematics, Physics
- 2006

The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space with p being a prime and r a positive integer, are shown to be qualitatively…

Equiangular Vectors Approach to Mutually Unbiased Bases

- Mathematics, Computer ScienceEntropy
- 2013

This work shows how to transform the problem of finding d + 1 mutually unbiased bases in the d-dimensional space Cd (with a modulus for the inner product) into the one ofFinding d(d+1) vectors in thed2-dimensionalspace Cd2 (without a moduli for the outer product).

Galois unitaries, mutually unbiased bases, and mub-balanced states

- Computer Science, MathematicsQuantum Inf. Comput.
- 2015

It is shown that there exist transformations that cycle through all the bases in all dimensions d = pn where p is an odd prime and the exponent n is odd, and it is conjecture that this construction yields all such states in odd prime power dimension.

SU2 Nonstandard Bases: Case of Mutually Unbiased Bases

- Mathematics
- 2007

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU2 corresponding to an ir- reducible representation of…

SU(2) nonstandard bases: the case of mutually unbiased bases

- Physics, Mathematics
- 2007

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU(2) corresponding to an irreducible representation of…

Affine Constellations Without Mutually Unbiased Counterparts

- Mathematics, Physics
- 2010

It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and…

## References

SHOWING 1-10 OF 24 REFERENCES

Constructions of Mutually Unbiased Bases

- Mathematics, Computer ScienceInternational Conference on Finite Fields and Applications
- 2003

This work gives a simplified proof of this fact based on the estimation of exponential sums that extremal sets containing d+1 mutually unbiased bases are known to exist.

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

- Physics, 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…

Mutually unbiased bases and finite projective planes

- Mathematics, Physics
- 2004

It is conjectured that the question of the existence of a set of d +1 mutu ally unbiased bases in a d-dimensional Hilbert space if d differs from a power of ap rimenumber is intimately linked with…

If 1=2+3, then 1=2.3: Bell states, finite groups, and mutually unbiased bases, a unifying approach

- Physics, Mathematics
- 2004

We study the relationship between Bell states, finite groups and complete sets of bases. We show how to obtain a set of N+1 bases in which Bell states are invariant.
They generalize the X, Y and Z…

Discrete phase space based on finite fields

- Physics
- 2004

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete…

A New Proof for the Existence of Mutually Unbiased Bases

- Mathematics, Computer ScienceAlgorithmica
- 2002

A constructive proof of the existence of mutually biased bases for dimensions that are powers of primes is presented and it is proved that in any dimension d the number of mutually unbiased bases is at most d+1.

Hilbert-Schmidt volume of the set of mixed quantum states

- Mathematics
- 2003

We compute the volume of the convex (N 2 −1)-dimensional set MN of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyperarea of the boundary of this set is also found and…