发表机构
School of Computer Science and Technology, Dongguan University of Technology; Department of Applied Mathematics, The Hong Kong Polytechnic University（东莞理工学院计算机科学与技术学院; 香港理工大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维纠缠，提出多尺度施密特谱界，连接几何度量与施密特数见证者，推导尖锐上界、优化解及混合态下界，并给出定量校准。
AI 中文摘要
高维双粒子纠缠取决于概率在施密特谱上的分布方式，而单一参考态保真度仅解析一个谱尺度。我们发展了连接几何度量与定量施密特数见证者的多尺度施密特谱界。对于纯态施密特谱的任意划分，若干嵌套的维达尔尾部确定块质量。我们推导了相应归一化核范数坐标的尖锐上界，并证明当且仅当谱在每个块内均匀时等式成立。细化划分在新增尾部数据能区分不等块均值时，给出更紧界的单调层级。我们还全局求解了一个松弛的加权多尺度优化：一个标量参数唯一确定其全支撑优化器及非平凡分支上的显式值。以已建立的单尾保真度-资源曲线为基线，我们获得了精确保真度等式细化、混合态的凸包下界以及施密特数见证者的定量校准。一个更高阶尾部关系进一步用维达尔尾部界定凸包扩展负性，并识别纯态等式谱。泄漏感知和联合置信度公式说明了这些界如何用于不完整数据。多步尾部需要块分辨或独立认证的谱信息；它们不由一个投影仪期望值决定。
英文摘要
High-dimensional bipartite entanglement depends on how probability is distributed across the Schmidt spectrum, whereas a single reference-state fidelity resolves only one spectral scale. We develop multiscale Schmidt-spectrum bounds that connect geometric measures with quantitative Schmidt-number witnesses. For any partition of a pure-state Schmidt spectrum, several nested Vidal tails determine block masses. We derive the sharp upper boundary of the associated normalized nuclear-norm coordinate and show that equality holds if and only if the spectrum is uniform within each block. Refining the partition gives a monotone hierarchy of tighter bounds whenever the added tail data distinguish unequal block means. We also solve a relaxed weighted multiscale optimization globally: one scalar parameter specifies its unique full-support optimizer and explicit value on the nontrivial branch. Using the established single-tail fidelity--resource curve as a baseline, we obtain exact-fidelity equality refinements, convex-roof lower bounds for mixed states, and quantitative calibrations of Schmidt-number witnesses. A higher-tail relation further bounds convex-roof extended negativity in terms of Vidal tails and identifies the pure-state equality spectra. Leakage-aware and joint-confidence formulations state how these bounds can be used with incomplete data. Multistep tails require block-resolved or independently certified spectral information; they are not determined by one projector expectation.
Comments36 pages
