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Operator Schmidt Rank and the Geometry of Schmidt Profiles
Abstract
The main result of the paper states that if 0≠ v∈ℂm⊗ℂn, d=min{m,n}, and p=SR(v), then for every 1≤ q≤ d, min{OSR(U): l U∈𝕄m⊗𝕄n is unitary, SR(Uv)=q } = ⌈max{p/q,q/p}⌉ . Here OSR(U) denotes the operator Schmidt rank of U. Furthermore, if U∈𝕄m⊗𝕄n is unitary and OSR(U)≤ k, then, for every unit vector x∈ℂm⊗ℂn and every 1≤ r≤ d, Fmin{kr,d}(Ux)≥ Fr(x) and Fmin{kr,d}(x)≥ Fr(Ux), where Fr(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.
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BibTeX
@misc{kian2026operator,
title = {{Operator Schmidt Rank and the Geometry of Schmidt Profiles}},
author = {Mohsen Kian and Mohammad Sal Moslehian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09794},
eprint = {2610.09794},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09794},
}
RIS
TY - UNPB
TI - Operator Schmidt Rank and the Geometry of Schmidt Profiles
AU - Mohsen Kian
AU - Mohammad Sal Moslehian
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09794
UR - https://arxiv.org/abs/2610.09794
AB - The main result of the paper states that if 0≠ v∈ℂ^m⊗ℂ^n, d=min{m,n}, and p=SR(v), then for every 1≤ q≤ d, min{OSR(U): l U∈𝕄_m⊗𝕄_n is unitary, SR(Uv)=q } = ⌈max{p/q,q/p}⌉ . Here OSR(U) denotes the operator Schmidt rank of U. Furthermore, if U∈𝕄_m⊗𝕄_n is unitary and OSR(U)≤ k, then, for every unit vector x∈ℂ^m⊗ℂ^n and every 1≤ r≤ d, F_min{kr,d}(Ux)≥ F_r(x) and F_min{kr,d}(x)≥ F_r(Ux), 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.
ER -