Purity maxima, absolute separability, and spectral volumes of absolutely PPT states

Abstract

For $m\otimes n$ systems with $n\ge m\ge3$, we determine the exact maximum purity of absolutely positive partial transpose (APPT) states and the unique maximizing ordered spectrum when $n\ge2m^2-4$ for even $m$ or $n\ge2m^2-m-4$ for odd $m$. Under these dimension assumptions, absolutely separable states have the same maximum purity, attained at the same unique ordered spectrum. The proof uses an outer polytope defined by two linear head--tail inequalities, each comparing a sum of the largest eigenvalues with a sum of the smallest. Under explicit multiplicity assumptions, we derive an exact matrix criterion for APPT membership of full-rank spectra with three distinct eigenvalues. For every $m\ge10$, this criterion gives explicit APPT boundary spectra at $n=4m-2$ and $n=4m$ that disprove both cases of the maximum purity conjecture of Ahiable, Kothakonda, and Winter. We also determine the exact completely absolutely separable (CAS) purity maximum and its unique maximizing ordered spectrum in every bipartite dimension. We prove that the inner APPT polytope consists entirely of absolutely separable spectra. The resulting linear sufficient condition for absolute separability strictly improves the reduction-map and eigenvalue-ratio tests. Finally, we compute the exact relative volumes of the CAS set and an outer polytope defined by a single head--tail inequality. Under the uniform measure on spectra, the CAS, absolutely separable, and APPT volumes have the same asymptotic order $\sqrt n\,e^{-κ_m n}$ for every fixed $m\ge2$, with explicit $κ_m>0$.