Tight informationally complete quantum measurements

@article{Scott2006TightIC,
  title={Tight informationally complete quantum measurements},
  author={A. J. Scott},
  journal={Journal of Physics A},
  year={2006},
  volume={39},
  pages={13507-13530}
}
  • A. J. Scott
  • Published 7 April 2006
  • Mathematics
  • Journal of Physics A
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 states. These measures are distinguished by an exceptionally simple state-reconstruction formula which allows 'painless' quantum state tomography. Complete sets of mutually unbiased bases and symmetric informationally complete positive-operator-valued measures are both members of this class, the latter… 
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.
Experimental Realization of Quantum Tomography of Photonic Qudits via Symmetric Informationally Complete Positive Operator-Valued Measures
Symmetric informationally complete positive operator-valued measures provide efficient quantum state tomography in any finite dimension. In this work, we implement state tomography using symmetric
Analysis and Synthesis of Minimal Informationally Complete Quantum Measurements.
Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. We establish general properties of MICs, explore
Experimental characterization of qutrits using symmetric informationally complete positive operator-valued measurements
Generalized quantum measurements [also known as positive operator-valued measures (POVMs)] are of great importance in quantum information and quantum foundations but are often difficult to perform.
Compressively Certifying Quantum Measurements
We introduce a reliable compressive procedure to uniquely characterize any given low-rank quantum measurement using a minimal set of probe states that is based solely on data collected from the
Discrete Wigner functions from informationally complete quantum measurements
Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than
Experimental characterization of qutrits using symmetric, informationally complete positive operator-valued measures
Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We
Triply Positive Matrices and Quantum Measurements Motivated by QBism
We study a class of quantum measurements that furnish probabilistic representations of finite-dimensional quantum theory. The Gram matrices associated with these Minimal Informationally Complete
Mutually Unbiased Product Bases
A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are
...
...

References

SHOWING 1-10 OF 83 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.
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 that
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
Minimal Informationally Complete Measurements for Pure States
We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a
Informationally complete measurements on bipartite quantum systems: Comparing local with global measurements
Informationally complete measurements allow the estimation of expectation values of any operator on a quantum system, by changing only the data processing of the measurement outcomes. In particular,
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.
Mutually unbiased bases are complex projective 2-designs
TLDR
It is demonstrated that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0, 1/d}.
Optimal tight frames and quantum measurement
TLDR
The well-known canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling and frame-theoretical analogs of various quantum-mechanical concepts and results are developed.
Reexamination of optimal quantum state estimation of pure states (5 pages)
A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by Hayashi in
A de Finetti representation for finite symmetric quantum states
Consider a symmetric quantum state on an n-fold product space, that is, the state is invariant under permutations of the n subsystems. We show that, conditioned on the outcomes of an informationally
...
...