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PPT states of almost maximal Schmidt number
Abstract
We construct PPT states on ℂm ⊗ℂn that have Schmidt number asymptotically approaching the smaller local dimension. More specifically, we construct a PPT state with Schmidt number at least ⌈ (m + n − ((m − n)2 + 4(m + n − 1))1/2)/2 ⌉. In the case of equal local dimensions (m = n), this becomes n − ⌊(2n − 1)1/2⌋, far exceeding the previous constructions which achieved n/2 + O(1). In the case of unequal local dimensions, our result shows that there exists a PPT state on ℂn ⊗ ℂ3n−4 with Schmidt number at least n−1.
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BibTeX
@misc{johnston2026states,
title = {{PPT states of almost maximal Schmidt number}},
author = {Nathaniel Johnston and Benjamin Lovitz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11849},
eprint = {2609.11849},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11849},
}
RIS
TY - UNPB
TI - PPT states of almost maximal Schmidt number
AU - Nathaniel Johnston
AU - Benjamin Lovitz
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11849
UR - https://arxiv.org/abs/2609.11849
AB - We construct PPT states on ℂ^m ⊗ℂ^n that have Schmidt number asymptotically approaching the smaller local dimension. More specifically, we construct a PPT state with Schmidt number at least ⌈ (m + n − ((m − n)^2 + 4(m + n − 1))^1/2)/2 ⌉. In the case of equal local dimensions (m = n), this becomes n − ⌊(2n − 1)^1/2⌋, far exceeding the previous constructions which achieved n/2 + O(1). In the case of unequal local dimensions, our result shows that there exists a PPT state on ℂ^n ⊗ ℂ^3n−4 with Schmidt number at least n−1.
ER -