The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States
Abstract
For a bipartite state ρ, information about the spectrum of its partial transpose ρΓB can be inferred from measurements on multiple copies of ρ, without full state tomography. This raises a natural question: which eigenvalue lists can arise as spec(ρΓB) for a density operator ρ? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as spec(ρΓB) by some PPT state ρ, whereas an ordered candidate eigenvalue list $(x,y,z,-q)$, with x≥ y≥ z≥0, q>0, and x+y+z−q=1, is realized by an NPT state iff q≤ y and qy≤ xz. Sufficiency in the latter case is established by an explicit X state whose quantum steering ellipsoid has center c=(y−q)/(1−z) and normalized volume V/Vmax(c)=qy/(xz), providing a geometric interpretation of the inequalities q≤ y and qy≤ xz as the allowed ellipsoid-center region and the fixed-center volume bound. Beyond this geometric picture, the two-qubit inverse theorem also yields exact negativity bounds from the two lowest nontrivial PT moments. Given fixed values of p2=Tr[(ρΓB)2] and p3=Tr[(ρΓB)3], we determine the exact minimum and maximum negativity over all two-qubit states subject to these moment constraints. When no PPT state is consistent with the pair (p2,p3), the minimum is attained either at x=y or qy=xz, while the maximum is attained either at y=z or q=y. Finally, we show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.
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BibTeX
@misc{dou2026inverse,
title = {{The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States}},
author = {Ruoting Dou and Shengjun Wu and Zeng-Bing Chen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29068},
eprint = {2608.29068},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29068},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States
AU - Ruoting Dou
AU - Shengjun Wu
AU - Zeng-Bing Chen
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2608.29068
UR - https://arxiv.org/abs/2608.29068
AB - For a bipartite state ρ, information about the spectrum of its partial transpose ρ^Γ_B can be inferred from measurements on multiple copies of ρ, without full state tomography. This raises a natural question: which eigenvalue lists can arise as spec(ρ^Γ_B) for a density operator ρ? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as spec(ρ^Γ_B) by some PPT state ρ, whereas an ordered candidate eigenvalue list $(x,y,z,-q)$, with x≥ y≥ z≥0, q>0, and x+y+z−q=1, is realized by an NPT state iff q≤ y and qy≤ xz. Sufficiency in the latter case is established by an explicit X state whose quantum steering ellipsoid has center c=(y−q)/(1−z) and normalized volume V/V_max(c)=qy/(xz), providing a geometric interpretation of the inequalities q≤ y and qy≤ xz as the allowed ellipsoid-center region and the fixed-center volume bound. Beyond this geometric picture, the two-qubit inverse theorem also yields exact negativity bounds from the two lowest nontrivial PT moments. Given fixed values of p_2=Tr[(ρ^Γ_B)^2] and p_3=Tr[(ρ^Γ_B)^3], we determine the exact minimum and maximum negativity over all two-qubit states subject to these moment constraints. When no PPT state is consistent with the pair (p_2,p_3), the minimum is attained either at x=y or qy=xz, while the maximum is attained either at y=z or q=y. Finally, we show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.
ER -