Symmetric informationally complete measurements of arbitrary rank

@article{Appleby2007SymmetricIC,
  title={Symmetric informationally complete measurements of arbitrary rank},
  author={D. M. Appleby},
  journal={Optics and Spectroscopy},
  year={2007},
  volume={103},
  pages={416-428}
}
  • D. M. Appleby
  • Published 2007
  • Mathematics, Physics
  • Optics and Spectroscopy
There has been much interest in so-called SIC-POVMs, i.e., rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs that are symmetric and informationally complete, but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM that is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of… Expand
Construction of all general symmetric informationally complete measurements
TLDR
The set of all general SIC-POVMs is constructed and it is shown that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general S IC POVM and vice versa. Expand
Two constructions of approximately symmetric informationally complete positive operator-valued measures
Symmetric informationally complete positive operator-valued measures (SIC-POVMs) have many applications in quantum information. However, it is not easy to construct SIC-POVMs and there are only a fewExpand
The Lie Algebraic Significance of Symmetric Informationally Complete Measurements
Examples of symmetric informationally complete positive operator-valued measures (SIC-POVMs) have been constructed in every dimension ⩽67. However, it remains an open question whether they exist inExpand
Galois automorphisms of a symmetric measurement
TLDR
The Galois group of SICs covariant with respect to the Weyl-Heisenberg group is examined and a list of nine conjectures concerning its structure are proposed, representing a considerable strengthening of the theorems actually proved. Expand
Group theoretic, lie algebraic and Jordan algebraic formulations of the sic existence problem
TLDR
The purpose of this paper is to show that the SIC existence problem is equivalent to three other, on the face of it quite different problems, although it is still not clear whether these reformulations of the problem will make it more tractable. Expand
Studies of symmetries that give special quantum states the "right to exist"
In this thesis we study symmetric structures in Hilbert spaces known as symmetric informationally complete positive operator-valued measures (SIC-POVMs), mutually unbiased bases (MUBs), andExpand
Notes on general SIC-POVMs
An unavoidable task in quantum information processing is how to obtain data about the state of an individual system by suitable measurements. Informationally complete measurements are relevant inExpand
Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States
TLDR
It is shown that if SIC-sets exist, they are as close to being an orthonormal basis for the space of density operators as possible and in prime dimensions, the standard construction for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant Sic-sets are intimately related. Expand
Mutually unbiased probability-operator measurements
We generalize the concept of unbiasedness from bases to measurements. We show that mutually unbiased (MU) measurements, which are not necessarily von Neumann measurements, exist in allExpand
Tight Frames, Hadamard Matrices and Zauner's Conjecture
We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures:Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 54 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
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–HeisenbergExpand
Mutually Unbiased Bases, Generalized Spin Matrices and Separability
Abstract A collection of orthonormal bases for a complex d-dimensional Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the innerExpand
On the quantumness of a hilbert space
  • C. Fuchs
  • Mathematics, Computer Science
  • Quantum Inf. Comput.
  • 2004
TLDR
It is established that the accessible fidelity for symmetric informationallycomplete signal ensembles is equal to the quantumness and any measurement consisting of rank-one POVM elements is an optimal measurement. Expand
Informationally complete measurements and group representation
Informationally complete measurements on a quantum system allow one to estimate the expectation value of any arbitrary operator by just averaging functions of the experimental outcomes. We show thatExpand
Mutually Unbiased Bases and the Complementarity Polytope
TLDR
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 to see if some values of N are geometrically singled out. Expand
Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
A parametrization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parametrization we find theExpand
Tomography of Quantum States in Small Dimensions
  • M. Grassl
  • Mathematics, Computer Science
  • Electron. Notes Discret. Math.
  • 2005
TLDR
This paper supports the conjecture that solutions for projective POVMs exist in any dimension by constructing explicit algebraic solutions in small dimensions d, in particular d = 12. Expand
Optimal state-determination by mutually unbiased measurements
For quantum systems having a finite number N of orthogonal states, we investigate a particular relation among different measurements, called “mutual unbiasedness,” which we show plays a special roleExpand
Informationally complete sets of physical quantities
The notion of informational completeness is formulated within the convex state (or operational) approach to statistical physical theories and employed to introduce a type of statistical metrics.Expand
...
1
2
3
4
5
...