算子 Schmidt 秩与 Schmidt 轮廓的几何性质

中文摘要 AI 辅助

本文的主要结果指出：若 $0\neq v\in\mathbb C^m\otimes\mathbb C^n$，$d=\min\{m,n\}$，且 $p=SR(v)$，则对每个 $1\le q\le d$，有 \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ 是酉的}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil. \end{equation*} 这里 $OSR(\mathbf U)$ 表示 $\mathbf U$ 的算子 Schmidt 秩。此外，若 $\mathbf U\in\mathbb M_m\otimes\mathbb M_n$ 是酉的且 $OSR(\mathbf U)\le k$，则对每个单位向量 $x\in\mathbb C^m\otimes\mathbb C^n$ 和每个 $1\le r\le d$，有 \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{且}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} 其中 $F_r(x)$ 表示 $x$ 的 $r$ 个最大平方 Schmidt 系数之和。此结果强于秩可达性，因为它控制了整个累积 Schmidt 轮廓。我们还建立了一个优序关系，并利用它获得了在具有有界算子 Schmidt 秩的酉变换下 Schatten 范数畸变和 Rényi 熵变化的尖锐界。

英文摘要

The main result of the paper states that if $0\neq v\in\mathbb C^m\otimes\mathbb C^n$, $d=\min\{m,n\}$, and $p=SR(v)$, then for every $1\le q\le d$, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here $OSR(\mathbf U)$ denotes the operator Schmidt rank of $\mathbf U$. Furthermore, if $\mathbf U\in\mathbb M_m\otimes\mathbb M_n$ is unitary and $OSR(\mathbf U)\le k$, then, for every unit vector $x\in\mathbb C^m\otimes\mathbb C^n$ and every $1\le r\le d$, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.

发表机构

• University of Bojnord（博季努尔德大学）
• Ferdowsi University of Mashhad（马什哈德菲尔多西大学）

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