SIC-POVMs and the Knaster's Conjecture

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 (n²-1) of the matrices have the same value of Tr(ρᵏ). 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(ρ³) on the Bloch sphere of $3$ and $4$ dimensional Hilbert spaces. In the $3$-dimensional Hilbert space, we generate 10⁴ generalized SIC-POVMs for randomly chosen Tr(ρ³) values such that all the elements are equivalent up to unitary transformations.