# 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…

## 208 Citations

Symmetric Informationally Complete Positive Operator Valued Measures

- Mathematics
- 2017

We consider the question of the existence of d2 equiangular lines in d-dimensional complex space Cd. In physics, such a set of equiangular lines is called a symmetric, informationally complete…

Constructing symmetric informationally complete positive-operator-valued measures in Bloch space

- Mathematics, Physics
- 2012

The Lie Algebraic Significance of Symmetric Informationally Complete Measurements

- Mathematics
- 2011

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 in…

Two new constructions of approximately symmetric informationally complete positive operator-valued measures

- MathematicsInternational Journal of Quantum Information
- 2019

A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in [Formula: see text] consisting of [Formula: see text] positive operators of rank one such that all of…

Two constructions of approximately symmetric informationally complete positive operator-valued measures

- Mathematics, Computer Science
- 2017

This paper uses character sums over finite fields to present two constructions of ASIC-POVMs and shows that there are some classes of infinite families of ASIC -POVs by using some special functions over infinite fields.

SIC-POVMS AND THE STARK CONJECTURES

- Mathematics
- 2018

The existence of a set of d pairwise equiangular complex lines (a SIC-POVM) in ddimensional Hilbert space is currently known only for a finite set of dimensions d. We prove that, if there exists a…

Geometric and Information-Theoretic Properties of the Hoggar Lines

- Computer Science
- 2016

Investigating the shape of this representation of state space leads to an intriguing link between the questions of real and of complex equiangular lines and relations between quantum information theory and mathematical topics like octonionic integers and the 28 bitangents to a quartic curve.

On symmetric decompositions of positive operators

- Mathematics
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Inspired by some problems in Quantum Information Theory, we present some results concerning decompositions of positive operators acting on finite dimensional Hilbert spaces. We focus on…

Fibonacci-Lucas SIC-POVMs

- Mathematics
- 2017

We present a conjectured family of symmetric informationally complete positive operator valued measures which have an additional symmetry group whose size is growing with the dimension. The symmetry…

Conical Designs and Categorical Jordan Algebraic Post-Quantum Theories

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Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in…

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