SIC-POVMs and the Knaster's Conjecture
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909211158380544 |
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| author | Samuel, S. B. Gedik, Z. |
| author_facet | Samuel, S. B. Gedik, Z. |
| contents | 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 $Tr(ρ^k)$. 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 $Tr(ρ^3)$ 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 $Tr(ρ^3)$ values such that all the elements are equivalent up to unitary transformations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_16733 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | SIC-POVMs and the Knaster's Conjecture Samuel, S. B. Gedik, Z. Quantum Physics 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 $Tr(ρ^k)$. 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 $Tr(ρ^3)$ 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 $Tr(ρ^3)$ values such that all the elements are equivalent up to unitary transformations. |
| title | SIC-POVMs and the Knaster's Conjecture |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2405.16733 |