$d \times d$ 相空间上离散维格纳表示的完整参数化与参数空间拓扑
Complete parameterization and parameter-space topology of discrete Wigner representations on $d \times d$ phase space
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中文总结 AI 辅助
本文完整参数化并分类了有限维系统上满足厄米性、单位迹、正交性和协变性的离散维格纳函数，给出了奇偶维度的参数空间拓扑，并区分了有效性固有自由度与边际及位移代数约束的选择。
中文摘要 AI 辅助
有限维系统的环形离散维格纳函数（DWF）并非唯一。我们对标记的单量子比特 $d\times d$ 表示进行分类，这些表示的相点算符是厄米的、单位迹的、希尔伯特-施密特正交的，并且是外尔-海森堡协变的。我们先前建立的模板定理将此族表示为加倍后的 $2d\times2d$ 父表示的下降。在此，我们显式地求解其投影模板可容许性条件。在辛-傅里叶表示中，可容许性固定了模量，留下 $d\times d$ 基元胞上的相位数据，满足依赖于奇偶性的扭曲奇函数关系。这给出了参数空间：对于奇数 $d$ 为 $(S^1)^{(d^2-1)/2}$，对于偶数 $d$ 为 $(S^1)^{(d^2-4)/2}\times\mathbb{Z}_2^3$。规范的水平与垂直边际分别将这些空间缩减为 $(S^1)^{(d-1)^2/2}$ 和 $(S^1)^{d(d-2)/2}\times\mathbb{Z}_2$。相同的相位数据参数化了加倍外尔-海森堡位移算符的保持有效性的重相位。要求这些算符的阶整除希尔伯特空间维数 $d$，则将每个 $S^1$ 因子替换为离散的 $\mathbb{Z}_d$ 因子。相关的辛-傅里叶特征函数编码相同的算符信息，其逐点模量与有效的模板约定无关。这种分类将DWF有效性固有的自由度与由边际和位移代数要求所选择的自由度区分开来，并使奇数维与偶数维之间的结构差异变得明确。
英文摘要
Toroidal discrete Wigner functions (DWFs) for finite-dimensional systems are non-unique. We completely parameterize and topologically classify the labeled single-qudit ${d\times d}$ representations whose phase-point operators are Hermitian, unit-trace, Hilbert--Schmidt orthogonal, and Weyl--Heisenberg covariant, determining the independent parameter freedom in every dimension. Our previously established stencil theorem expresses this family as descents of a doubled ${2d\times2d}$ parent Wigner representation. Here, we solve its projected-stencil admissibility conditions explicitly to obtain the full parameter space of this family. In the symplectic-Fourier representation, admissibility fixes the modulus, leaving phase data on a ${d\times d}$ base cell satisfying a parity-dependent twisted-oddness relation. This gives the parameter space ${(S^1)^{(d^2-1)/2}}$ for odd ${d}$ and ${(S^1)^{(d^2-4)/2}\times\mathbb{Z}_{2}^{3}}$ for even ${d}$. Canonical horizontal and vertical marginals reduce these spaces to ${(S^1)^{(d-1)^2/2}}$ and ${(S^1)^{d\,(d-2)/2}\times\mathbb{Z}_{2}}$, respectively. The same phase data parameterize validity-preserving rephasings of the doubled Weyl--Heisenberg displacement operators. Requiring these operators to have order dividing the Hilbert-space dimension ${d}$ replaces each ${S^1}$ factor by a discrete ${\mathbb{Z}_{d}}$ factor. The associated symplectic-Fourier characteristic functions encode the same operator information, with pointwise magnitudes independent of the valid stencil convention and hence providing convention-independent information about the represented operator within this family. This classification separates the freedom intrinsic to DWF validity from that selected by marginal and displacement-algebra requirements and makes explicit the structural distinction between odd and even dimensions.
发表机构
- RMIT University（皇家墨尔本理工大学）
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