On Obtaining New MUBs by Finding Points on Complete Intersection Varieties over $\mathbb{R}$
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| Format: | Preprint |
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2025
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| author | Banerjee, Arindam Das, Kanoy Kumar Kumar, Ajeet Kumar, Rakesh Maitra, Subhamoy |
| author_facet | Banerjee, Arindam Das, Kanoy Kumar Kumar, Ajeet Kumar, Rakesh Maitra, Subhamoy |
| contents | Mutually Unbiased Bases (MUBs) are closely connected with quantum physics, and the structure has a rich mathematical background. We provide equivalent criteria for extending a set of MUBs for $C^n$ by studying real points of a certain affine algebraic variety. This variety comes from the relations that determine the extendability of a system of MUBs. Finally, we show that some part of this variety gives rise to complete intersection domains. Further, we show that there is a one-to-one correspondence between MUBs and the maximal commuting classes (bases) of orthogonal normal matrices in $\mathcal M_n({\mathbb{C}})$. It means that for $m$ MUBs in $C^n$, there are $m$ commuting classes, each consisting of $n$ commuting orthogonal normal matrices and the existence of maximal commuting basis for $\mathcal M_n({\mathbb{C}})$ ensures the complete set of MUBs in $\mathcal M_n({\mathbb{C}})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Obtaining New MUBs by Finding Points on Complete Intersection Varieties over $\mathbb{R}$ Banerjee, Arindam Das, Kanoy Kumar Kumar, Ajeet Kumar, Rakesh Maitra, Subhamoy Discrete Mathematics Mutually Unbiased Bases (MUBs) are closely connected with quantum physics, and the structure has a rich mathematical background. We provide equivalent criteria for extending a set of MUBs for $C^n$ by studying real points of a certain affine algebraic variety. This variety comes from the relations that determine the extendability of a system of MUBs. Finally, we show that some part of this variety gives rise to complete intersection domains. Further, we show that there is a one-to-one correspondence between MUBs and the maximal commuting classes (bases) of orthogonal normal matrices in $\mathcal M_n({\mathbb{C}})$. It means that for $m$ MUBs in $C^n$, there are $m$ commuting classes, each consisting of $n$ commuting orthogonal normal matrices and the existence of maximal commuting basis for $\mathcal M_n({\mathbb{C}})$ ensures the complete set of MUBs in $\mathcal M_n({\mathbb{C}})$. |
| title | On Obtaining New MUBs by Finding Points on Complete Intersection Varieties over $\mathbb{R}$ |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2507.02492 |