Mutually Unbiased Bases and the Complementarity Polytope

  title={Mutually Unbiased Bases and the Complementarity Polytope},
  author={Ingemar Bengtsson and {\AA}sa Ericsson},
  journal={Open Systems \& Information Dynamics},
  • I. Bengtsson, Åsa Ericsson
  • Published 15 October 2004
  • Mathematics, Computer Science, Physics
  • Open Systems & Information Dynamics
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… 
A gap for the maximum number of mutually unbiased bases
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
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
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
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
  • M. Kibler
  • Mathematics, Computer Science
  • 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
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
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
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
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


Mutually Unbiased Bases, Generalized Spin Matrices and Separability
Constructions of Mutually Unbiased Bases
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
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
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
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
Optimal state-determination by mutually unbiased measurements
If 1=2+3, then 1=2.3: Bell states, finite groups, and mutually unbiased bases, a unifying approach
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
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
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
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