The geometry of absolute separability and other convex matrix properties from spectrum

We investigate the geometric structure of the set of spectra of bipartite absolute separable states (ASEP m,n) and absolute positive partial transpose states (APPT m,n), i.e., bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that APPT m,n is a spectrahedron for all m at most n: in particular, all its faces are exposed. While ASEP 2,n is a spectrahedron, too, for ASEP m,n we can in general only show that it is a semialgebraic set for all m,n at least 3. Furthermore, we provide a complete characterization of the faces and extreme points of APPT m,n and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension mn-m-1. In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of APPT m,n via an inscribed polytope P m,n and conjecture that the maximal purity of APPT m,n (along with its spectra) coincides with the polytope for arbitrary dimensions except when m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of APPT m,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope P m,n as the local system dimension n increases. Finally, we show that the relative spectral volume of APPT m,n decays exponentially in n by a constant multiplicative factor of the relative volume of the inscribed polytope P m,n.