Curator's Take
AI Commentary
This article settles a long‑standing question by giving a complete description of which eigenvalue lists can arise as the partial transpose of any two‑qubit state, distinguishing exactly when they correspond to separable (PPT) versus entangled (NPT) states. By linking the algebraic conditions on the spectrum to the geometry of quantum steering ellipsoids, it provides a concrete way to infer entanglement measures such as negativity from just a few PT moments—information that can be extracted experimentally without full state tomography. The result sharpens our toolbox for hardware‑level verification and sensing protocols, although extending the framework beyond two qubits will require further work.
— Mark Eatherly
Summary
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.