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Quantum multiresolution analysis via fast Fourier transforms on Heisenbrg finite group

机译:Heisenbrg有限群上通过快速傅里叶变换进行的量子多分辨率分析

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We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW_(2N+1) (Z_(m~n)~N, Z_(m~n)~N, Z_(m~n)) over the ring Z_(m~n). This group is the discrete version of the real Heisenberg group HW_(2N+1) (R~N,R~N,R), where R is the real field. These functions have one dimensional and m~1, m~2,..., m~n-dimensions matrix valued spectral components (for irreducible representations of 'HW_(2N+1). The family of all 1D representations gives classical world world (CW). Various m~i-dimension representations (i = 1,2,..., n) map classical world (CW) into quantum worlds QW(m~i) of ith resolution (i = 1,2,..., n). Worlds QW(m~1) and QW(m~n) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multireso-lution procedure. We call this transform the natural quantum Fourier transform.
机译:我们研究有限Heisenberg群HW_(2N + 1)(Z_(m〜n)〜N,Z_(m〜n)〜N,Z_(m〜n)上的相空间函数(经典观测)的经典和量子谐波分析)在环Z_(m〜n)上。该组是实海森堡组HW_(2N + 1)(R〜N,R〜N,R)的离散形式,其中R是实场。这些函数具有一维和m〜1,m〜2,...,m〜n维矩阵值的频谱分量(用于'HW_(2N + 1)的不可约表示。所有一维表示的族提供了经典的世界世界(CW)。各种m〜i维表示(i = 1,2,...,n)将经典世界(CW)映射为第i个分辨率(i = 1,2 ,.)的量子世界QW(m〜i)。 ..,n)。世界QW(m〜1)和QW(m〜n)分别包含有关量子词的粗略信息和精细细节,在这种情况下,可以将海森堡群上的傅立叶变换视为Weyl量化多分辨率-解过程,我们称此变换为自然量子傅里叶变换。

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