Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay
We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the $2\otimes3$ system, yields, for every $mn\ge6$, an explicit upper bound on the purity of APPT states that is asymptotic to $4/(3mn)$. This improves the previous $2/(mn)$ upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every $2\otimes n$ system with $n\ge3$. For every $3\otimes n$ system with $n\ge3$, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every $2\otimes n$ system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume $a_n$ whose ratio is less than $4$ and tends to $3$. Consequently, $a_n=Θ\!\left(\sqrt n(4/27)^n\right)$, and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate $\ln(27/4)$.