SIC-POVMs and the Knaster's Conjecture

• URL: http://arxiv.org/abs/2405.16733v1
• Date: Mon, 27 May 2024 00:31:03 GMT
• Title: SIC-POVMs and the Knaster's Conjecture
• Authors: S. B. Samuel, Z. Gedik,
• Abstract summary: We use the Bloch sphere representation of SIC-POVMs to prove the Knaster's conjecture for the geometry of SIC-POVMs. We also prove the existence of a continuous family of generalized SIC-POVMs where $(n2-1)$ of the matrices have the same value of $Tr(rhok)$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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