SIC-POVMs and the Knaster's Conjecture

TL;DR

This paper proves the Knaster's conjecture for SIC-POVMs' geometry, demonstrates the existence of continuous families of generalized SIC-POVMs, and constructs numerous examples in 3- and 4-dimensional spaces.

Contribution

It provides a geometric proof of Knaster's conjecture for SIC-POVMs and constructs a large set of generalized SIC-POVMs with specific trace properties.

Findings

Proved Knaster's conjecture for SIC-POVMs geometry.

Established existence of continuous families of generalized SIC-POVMs.

Generated 10,000 generalized SIC-POVMs in 3D space with specific trace characteristics.

Abstract

Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) have been constructed in many dimensions using the Weyl-Heisenberg group. In the quantum information community, it is commonly believed that SCI-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. The Bloch sphere representation of SIC-POVMs allows for a general geometric description of the set of operators, where they form the vertices of a regular simplex oriented based on a continuous function. We use this perspective of the SIC-POVMs to prove the Knaster's conjecture for the geometry of SIC-POVMs and prove the existence of a continuous family of generalized SIC-POVMs where $(n^2-1)$ of the matrices have the same value of $Tr(\rho^k)$. Furthermore, by using numerical methods, we show that a regular simplex can be constructed such that all its vertices map to the same value of $Tr(\rho^3)$ on the Bloch sphere of $3$ and $4$ dimensional Hilbert spaces. In the $3$-dimensional Hilbert space, we generate $10^4$ generalized SIC-POVMs for randomly chosen $Tr(\rho^3)$ values such that all the elements are equivalent up to unitary transformations.