From SICs and MUBs to Eddington

Abstract

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.

12 pages; from the Festschrift for Tony Sudbery

References in corpus (3)

• Abstract configurations in algebraic geometry
• Discrete phase space and minimum-uncertainty states
• The Hesse pencil of plane cubic curves

Cited by in corpus (19)

• The SIC Question: History and State of Play
• Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments
• Quasiprobability representations of quantum mechanics with minimal negativity
• Super-symmetric informationally complete measurements
• Experimental study of optimal measurements for quantum state tomography
• Exploring the geometry of qutrit state space using symmetric informationally complete probabilities
• Linear Dependencies in Weyl-Heisenberg Orbits
• Nonlocality, steering and quantum state tomography in a single experiment
• Magic informationally complete POVMs with permutations
• Symmetric minimal quantum tomography by successive measurements
• Experimental proposal for symmetric minimal two-qubit state tomography
• Orbits of mutually unbiased bases
• Systems of Imprimitivity for the Clifford Group
• The Poincaré half-plane for informationally complete POVMs
• tqix: A toolbox for Quantum in X: Quantum measurement, quantum tomography, quantum metrology, and others
• Uncertainty relations for unified ($\alpha$,$\beta$)-relative entropy of coherence under mutually unbiased equiangular tight frames
• Constructions of complex equiangular lines from mutually unbiased bases
• A geometrical relation between symmetric operators and mutually unbiased operators
• Evidence for and against Zauner's MUB Conjecture in ${\mathbb{C}}^6$