Higher Abelian Quantum Double Models
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908542599954432 |
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| author | Flores, Jorge Acuña De Nittis, Giuseppe Espiro, Javier Lorca |
| author_facet | Flores, Jorge Acuña De Nittis, Giuseppe Espiro, Javier Lorca |
| contents | This paper focuses on the generalized version of the quantum double model on arbitrary $N$-dimensional simplicial complexes with finite local regularity. The core of our analysis is a detailed characterization of the frustration-free ground state space $\mathrm{FG}_{\mathrm{QDM}}(\mathfrak{A})$. A central result is the construction of the algebra of logical operators $\mathfrak{A}_{\mathrm{log}} := \mathfrak{K}'/\mathfrak{J}$, where the redundancy ideal $\mathfrak{J}$ quotients out operators that act trivially on the ground state space. We prove a homeomorphism between the state space of $\mathfrak{A}_{\mathrm{log}}$ and $\mathrm{FG}_{\mathrm{QDM}}(\mathfrak{A})$, effectively classifying all frustration-free ground states. This logical algebra is shown to exhibit generalized Canonical Commutation Relations (CCR). When the relevant (co)homology groups are finite, $\mathfrak{A}_{\mathrm{log}}$ is isomorphic to $C(X_c) \otimes \mathcal{B}(\mathfrak{h}_q)$, revealing that the ground state space can encode $c$ classical bits and $q$ quantum bits (qubits), providing a precise measure of its information storage capacity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Abelian Quantum Double Models Flores, Jorge Acuña De Nittis, Giuseppe Espiro, Javier Lorca Mathematical Physics Primary: 81R15, Secondary:46L30, 47L40, 82B20 This paper focuses on the generalized version of the quantum double model on arbitrary $N$-dimensional simplicial complexes with finite local regularity. The core of our analysis is a detailed characterization of the frustration-free ground state space $\mathrm{FG}_{\mathrm{QDM}}(\mathfrak{A})$. A central result is the construction of the algebra of logical operators $\mathfrak{A}_{\mathrm{log}} := \mathfrak{K}'/\mathfrak{J}$, where the redundancy ideal $\mathfrak{J}$ quotients out operators that act trivially on the ground state space. We prove a homeomorphism between the state space of $\mathfrak{A}_{\mathrm{log}}$ and $\mathrm{FG}_{\mathrm{QDM}}(\mathfrak{A})$, effectively classifying all frustration-free ground states. This logical algebra is shown to exhibit generalized Canonical Commutation Relations (CCR). When the relevant (co)homology groups are finite, $\mathfrak{A}_{\mathrm{log}}$ is isomorphic to $C(X_c) \otimes \mathcal{B}(\mathfrak{h}_q)$, revealing that the ground state space can encode $c$ classical bits and $q$ quantum bits (qubits), providing a precise measure of its information storage capacity. |
| title | Higher Abelian Quantum Double Models |
| topic | Mathematical Physics Primary: 81R15, Secondary:46L30, 47L40, 82B20 |
| url | https://arxiv.org/abs/2509.12864 |