Construction of Metaplectic Representations of $SL_2(\mathbb{Z}_{2^n})$ and Twisted Magnetic Translations

• URL: http://arxiv.org/abs/2505.20983v2
• Date: Fri, 20 Jun 2025 13:03:53 GMT
• Title: Construction of Metaplectic Representations of $SL_2(\mathbb{Z}_{2^n})$ and Twisted Magnetic Translations
• Authors: Emmanuel Floratos, Kimon Manolas, Ioannis Tsohantjis,
• Abstract summary: Unitary metaplectic representations of the group $SL_2(mathbbZ_2n)$ are necessary to describe the evolution of $2n$-dimensional quantum systems.<n>It is shown that in order for the metaplectic property to be fulfilled, an increase in the dimensionality of the involved $n$-qubit Hilbert spaces, from $2n$ to $22n$, is necessary.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Unitary metaplectic representations of the group $SL_2(\mathbb{Z}_{2^n})$ are necessary to describe the evolution of $2^n$-dimensional quantum systems, such as systems involving $n$ qubits. It is shown that in order for the metaplectic property to be fulfilled, an increase in the dimensionality of the involved $n$-qubit Hilbert spaces, from $2^n$ to $2^{2n}$, is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $HW_{2^n} \otimes HW_{2^n}$. Comparisson with other approaches on this problem of the literature are discussed.

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