From SICs and MUBs to Eddington

2010/11/01 by Ingemar Bengtsson · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics #Group (periodic table) #Heisenberg group #Linguistics #Mathematics #Mutually unbiased bases #Philosophy #Physics #Prime (order theory) #Pure mathematics #Quantum mechanics #Statement (logic) #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1088/1742-6596/254/1/012007

published as J. Phys. Conf. Ser. 254 (2010) 012007 · 12 pages; from the Festschrift for Tony Sudbery

openalex publication_date 2010/11/01 · arxiv created 2011/03/10 · arxiv updated 2015/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• Discrete phase space and minimum-uncertainty states
• Symmetric informationally complete quantum measurements
• The Hesse pencil of plane cubic curves
• Geometric Quantum Mechanics
• Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States

Cited by

• Orbits of mutually unbiased bases

Related

• On SIC-POVMs and MUBs in Dimension 6
• SIC POVMs and Clifford groups in prime dimensions
• Some non-standard ways to generate SIC-POVMs in dimensions 2 and 3
• SICs: Extending the list of solutions
• SIC-POVMs and the Knaster's Conjecture