Symmetric informationally complete positive-operator-valued measures: A new computer study

@article{Scott2010SymmetricIC,
  title={Symmetric informationally complete positive-operator-valued measures: A new computer study},
  author={A. J. Scott and Markus Grassl},
  journal={Journal of Mathematical Physics},
  year={2010},
  volume={51},
  pages={042203}
}
We report on a new computer study of the existence of d2 equiangular lines in d complex dimensions. Such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory. We provide numerical solutions in all dimensions d≤67 and, moreover, a putatively complete list of Weyl–Heisenberg covariant solutions for d≤50. A symmetry analysis of this list leads to new… 
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References

SHOWING 1-10 OF 40 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.
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 quantum
The Frame Potential, on Average
TLDR
A Symmetric Informationally Complete Positive Operator Valued Measure consists of N2 equiangular unit vectors in an N dimensional Hilbert space and the frame potential becomes a function of a single fiducial vector.
Symmetric informationally complete–positive operator valued measures and the extended Clifford group
We describe the structure of the extended Clifford group [defined to be the group consisting of all operators, unitary and antiunitary, which normalize the generalized Pauli group (or Weyl–Heisenberg
Tomography of Quantum States in Small Dimensions
  • M. Grassl
  • Mathematics
    Electron. Notes Discret. Math.
  • 2005
Tight informationally complete quantum measurements
We introduce a class of informationally complete positive-operator-valued measures which are, in analogy with a tight frame, 'as close as possible' to orthonormal bases for the space of quantum
On SIC-POVMs in prime dimensions
The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs
On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states
TLDR
This work addresses the problem of constructing positive operator-valued measures in finite dimension n consisting of n2 operators of rank one which have an inner product close to uniform and presents two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed.
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
SIC‐POVMS and MUBS: Geometrical Relationships in Prime Dimension
The paper concerns Weyl‐Heisenberg covariant SIC‐POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When
...
1
2
3
4
...