Symmetric Informationally Complete Measurements of Arbitrary Rank

Abstract

There has been much interest in so-called SIC-POVMs: rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs which 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 which is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1 symmetric informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.

Contribution to proceedings of International Conference on Quantum Optics, Minsk, 2006

References in corpus (11)

• Symmetric Informationally Complete Quantum Measurements
• The Bloch Vector for N-Level Systems
• Unknown Quantum States: The Quantum de Finetti Representation
• Discrete phase space based on finite fields
• SIC-POVMs and the Extended Clifford Group
• Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
• Quantum computers in phase space
• Informationally complete measurements and groups representation
• Mutually Unbiased Bases and Finite Projective Planes
• Orbits of quantum states and geometry of Bloch vectors for $N$-level systems
• Wigner distributions for finite dimensional quantum systems: An algebraic approach

Cited by in corpus (41)

• The SIC Question: History and State of Play
• Introducing the Qplex: A Novel Arena for Quantum Theory
• Notes on general SIC-POVMs
• Universally Fisher-Symmetric Informationally Complete Measurements
• Quantum Conical Designs
• The Lie Algebraic Significance of Symmetric Informationally Complete Measurements
• Efficient Quantum State Sample Tomography with Basis-dependent Neural-networks
• Tight Frames, Hadamard Matrices and Zauner's Conjecture
• Entanglement detection via general SIC-POVMs
• Discrete Wigner Functions from Informationally Complete Quantum Measurements
• Quantum mechanics: The Bayesian theory generalised to the space of Hermitian matrices
• What are the minimal conditions required to define a SIC POVM?
• Negativity Bounds for Weyl-Heisenberg Quasiprobability Representations
• Entanglement and Designs
• Tomographic and Lie algebraic significance of generalized symmetric informationally complete measurements
• Entanglement criterion via general symmetric informationally complete measurements
• Construction of general symmetric-informationally-complete-positive-operator-valued measures by using a complete orthogonal basis
• Entanglement Witnesses Based on Symmetric Informationally Complete Measurements
• Total variance and invariant information in complementary measurements
• Systems of Imprimitivity for the Clifford Group
• Communication capacity of mixed quantum t designs
• Entropic uncertainty relations for SIC-POVMs and MUMs
• Uncertainty and complementarity relations based on generalized skew information
• Entanglement criterion via general symmetric informationally complete measurement
• Schmidt number criterion via general symmetric informationally complete measurements
• Is the SIC Outcome There When Nobody Looks?
• Detecting EPR steering via two classes of local measurements
• The Decompositions of Werner and Isotropic States
• Phase-space representations of SIC-POVM fiducial states
• Self-testing of semisymmetric informationally complete measurements in a qubit prepare-and-measure scenario
• Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces
• Computational Complexity and the Nature of Quantum Mechanics
• Maximal Orthoplectic Fusion Frames from Mutually Unbiased Bases and Block Designs
• Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories
• Quantum state tomography as a numerical optimization problem
• All classes of informationally complete symmetric measurements in finite dimensions
• On symmetric decompositions of positive operators
• Quantum Separability Criteria Based on Symmetric Measurements
• Enhanced separability criteria based on symmetric measurements
• Can QBism exist without Q? Morphophoric measurements in generalised probabilistic theories
• Informationally Overcomplete POVMs for Quantum State Estimation and Binary Detection