Multiscale Schmidt-Spectrum Bounds for High-Dimensional Entanglement: Geometric Measures and Schmidt-Number Witnesses
Abstract
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.
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BibTeX
@misc{xiong2026multiscale,
title = {{Multiscale Schmidt-Spectrum Bounds for High-Dimensional Entanglement: Geometric Measures and Schmidt-Number Witnesses}},
author = {Liang Xiong and Zhixiang Jin and Yanling Wang and Wei Chen and Hong Tao and Nung-sing Sze},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21209},
eprint = {2609.21209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21209},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Multiscale Schmidt-Spectrum Bounds for High-Dimensional Entanglement: Geometric Measures and Schmidt-Number Witnesses
AU - Liang Xiong
AU - Zhixiang Jin
AU - Yanling Wang
AU - Wei Chen
AU - Hong Tao
AU - Nung-sing Sze
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21209
UR - https://arxiv.org/abs/2609.21209
AB - 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.
ER -