Maximum purity of absolutely separable and absolutely PPT states in qutrit--qudit systems

Abstract

A bipartite state is absolutely positive partial transpose (APPT) if it remains PPT under every global unitary transformation, and absolutely separable if it remains separable. For every qutrit--qudit system $\mathbb{C}^3\otimes\mathbb{C}^n$, $n\ge3$, we prove that the maximum purity is the same for absolutely separable and APPT states and equals $\max\{(3n+8)/(3n+2)^2,\,3/(8n)\}$. The maximizing ordered spectrum is unique and contains exactly two distinct eigenvalues: the larger one has multiplicity one for $3\le n\le8$ and multiplicity $n$ for $n\ge9$. We also derive, in terms of the purity, an upper bound on the minimum partial-transpose eigenvalue achievable by global unitary transformations, and characterize all states for which the bound is attained. These results confirm the inscribed-polytope purity conjecture of Ahiable, Kothakonda, and Winter for all qutrit--qudit systems. They also give a lower bound, sharp over explicit purity ranges, on the global depolarizing noise needed for absolute separability or APPT.