Corpus ID: 16409243

On SIC-POVMs and MUBs in Dimension 6

@article{Grassl2004OnSA,
  title={On SIC-POVMs and MUBs in Dimension 6},
  author={M. Grassl},
  journal={arXiv: Quantum Physics},
  year={2004}
}
  • M. Grassl
  • Published 2004
  • Physics, Mathematics
  • arXiv: Quantum Physics
We provide a partial solution to the problem of constructing mutually unbiased bases (MUBs) and symmetric informationally complete POVMs (SIC-POVMs) in non-prime-power dimensions. An algebraic description of a SIC-POVM in dimension six is given. Furthermore it is shown that several sets of three mutually unbiased bases in dimension six are maximal, i.e., cannot be extended. 

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5
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References

SHOWING 1-10 OF 11 REFERENCES
Symmetric informationally complete quantum measurements
TLDR
It is conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim. Expand
A New Proof for the Existence of Mutually Unbiased Bases
TLDR
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. Expand
Quantum Measurements and Finite Geometry
A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantumExpand
Constructions of Mutually Unbiased Bases
TLDR
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. Expand
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 discreteExpand
The Magma Algebra System I: The User Language
TLDR
The MAGMA language is presented, the design principles and theoretical background are outlined, and the constructors for structures, maps, and sets are outlined. Expand
Elements of the Representation Theory of the Jacobi Group
The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functionsExpand
Gruppentheorie und Quantenmechanik
Leprobì eme de 36 officiers, Comptes Rendues de l'Assoc
  • Français Avanc. Sci. Naturel
  • 1900
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1
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