Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915867411873792 |
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| author | Damski, Bogdan S. Bistroń, Rafał Ponterio, Diego Czartowski, Jakub Życzkowski, Karol |
| author_facet | Damski, Bogdan S. Bistroń, Rafał Ponterio, Diego Czartowski, Jakub Życzkowski, Karol |
| contents | In this work we investigate discrete structures in product Hilbert spaces. For monopartite systems of size $d$ one relies on the Weyl-Heisenberg group $WH(d)$, while in the case of composite Hilbert spaces we identify designs covariant with respect to the product group, $[WH(p)]^{\otimes n}$. In analogy with magic -a quantity attaining its maximum for states fiducial with respect to $WH(d)$ -we introduce a similar notion of magick, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate $d$ $\textit{a priori}$ isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions $p^n$ with $p\ge 3$. The result for $p\ge 5$ extends the construction of Klappenecker and Rötteler, whereas for $p=3$ it is mathematically distinct and is based on Galois rings. The global maximum of magick for $d=2^3$ yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15550 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick Damski, Bogdan S. Bistroń, Rafał Ponterio, Diego Czartowski, Jakub Życzkowski, Karol Quantum Physics In this work we investigate discrete structures in product Hilbert spaces. For monopartite systems of size $d$ one relies on the Weyl-Heisenberg group $WH(d)$, while in the case of composite Hilbert spaces we identify designs covariant with respect to the product group, $[WH(p)]^{\otimes n}$. In analogy with magic -a quantity attaining its maximum for states fiducial with respect to $WH(d)$ -we introduce a similar notion of magick, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate $d$ $\textit{a priori}$ isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions $p^n$ with $p\ge 3$. The result for $p\ge 5$ extends the construction of Klappenecker and Rötteler, whereas for $p=3$ it is mathematically distinct and is based on Galois rings. The global maximum of magick for $d=2^3$ yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions. |
| title | Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2603.15550 |