A threshold for maximal Schmidt number from spectrum
Abstract
The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on ℂd ⊗ ℂd alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)-block positive operators, we prove that LSp⊆ ASNd−1 for p≤ ⌊ d2/2⌋, where LSp denotes the set of states whose largest eigenvalue does not exceed the sum of their p smallest eigenvalues, and ASNd−1 denotes the set of states whose Schmidt number remains at most d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASNd−1 in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASNd−1. The global unitary requirement is nevertheless essential: for every d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
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BibTeX
@misc{an2026threshold,
title = {{A threshold for maximal Schmidt number from spectrum}},
author = {Junhyeong An and Soojoon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34255},
eprint = {2609.34255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34255},
}
RIS
TY - UNPB
TI - A threshold for maximal Schmidt number from spectrum
AU - Junhyeong An
AU - Soojoon Lee
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2609.34255
UR - https://arxiv.org/abs/2609.34255
AB - The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on ℂ^d ⊗ ℂ^d alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)-block positive operators, we prove that LS_p⊆ ASN_d−1 for p≤ ⌊ d^2/2⌋, where LS_p denotes the set of states whose largest eigenvalue does not exceed the sum of their p smallest eigenvalues, and ASN_d−1 denotes the set of states whose Schmidt number remains at most d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASN_d−1 in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASN_d−1. The global unitary requirement is nevertheless essential: for every d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
ER -