Separable decompositions of 2xn states with operator Schmidt rank three
Abstract
We prove that every bipartite state ρ of a qubit coupled to an n-level system whose operator Schmidt rank equals three can be written as a mixture of rank(ρ) pure product states, and as a mixture of at most n+1 mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.
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BibTeX
@misc{vantalon2026separable,
title = {{Separable decompositions of 2xn states with operator Schmidt rank three}},
author = {Perrine Vantalon and Nicolas Macris},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39669},
eprint = {2609.39669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39669},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Separable decompositions of 2xn states with operator Schmidt rank three
AU - Perrine Vantalon
AU - Nicolas Macris
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39669
UR - https://arxiv.org/abs/2609.39669
AB - We prove that every bipartite state ρ of a qubit coupled to an n-level system whose operator Schmidt rank equals three can be written as a mixture of rank(ρ) pure product states, and as a mixture of at most n+1 mixed product states in general. The second result is tight in the sense that there exist states that cannot be decomposed with a smaller number of terms. Our proof gives constructive methods to obtain the decompositions. It involves eigendecompositions of suitable unitary dilations of an operator explicitly determined by the state. Our framework recovers previously known results for the length of decompositions into pure product states and is also able to deal with decompositions into mixed states. For pure product states, our decomposition is different from existing ones.
ER -