TL;DR
This survey explores the mathematical structures of MUBs and SICs, their geometric interpretations in low dimensions, and their connections to elliptic curves and the Heisenberg group, highlighting open problems in higher dimensions.
Contribution
It synthesizes existing knowledge on MUBs and SICs, emphasizing geometric and algebraic structures, and discusses the current understanding and challenges in dimensions 3 and 4.
Findings
MUBs relate to elliptic normal curves in prime dimensions
SICs are believed to be Heisenberg group orbits in all dimensions
Geometry of SICs in dimensions 3 and 4 is analyzed in detail
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.
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