The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States

• URL: http://arxiv.org/abs/2608.29068v1
• Date: Sat, 29 Aug 2026 05:44:57 GMT
• Title: The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States
• Abstract summary: We solve the inverse eigenvalue problem for two qubits.<n>Every nonnegative trace-one spectrum is realized as $operatornamespec(_B)$.<n>We show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.
• Score: 0.0611737116137921
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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 $\operatorname{spec}(ρ^{Γ_B})$ for a density operator $ρ$? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as $\operatorname{spec}(ρ^{Γ_B})$ by some PPT state $ρ$, whereas an ordered candidate eigenvalue list $(x,y,z,-q)$, with $x\ge y\ge z\ge0$, $q>0$, and $x+y+z-q=1$, is realized by an NPT state iff $q\le y$ and $qy\le xz$. 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 $q\le y$ and $qy\le xz$ 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.

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