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}
}
  • 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… 
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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.
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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.
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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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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
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TLDR
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