2026/08/04 by Jennifer Ahiable, Naga Bhavya Teja Kothakonda, Andreas Winter
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.MP
48 pages, 11 figures
arxiv created 2026/08/04 · arxiv updated 2026/08/05
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (ASEPm,n) and absolute positive partial transpose states (APPTm,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 APPTm,n is a spectrahedron for all m≤ n: in particular, all its faces are exposed. In contrast, while ASEP2,n is also a spectrahedron, we prove that in general ASEPm,n is a semialgebraic set for all m≤ n. Furthermore, we provide a complete characterization of the faces and extreme points of APPTm,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 APPTm,n via an inscribed polytope Pm,n and conjecture that the maximal purity of APPTm,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 APPTm,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope Pm,n as the local system dimension n increases. Finally, we show that the relative spectral volume of APPTm,n decays exponentially in n by a constant multiplicative factor of the relative volume of the inscribed polytope Pm,n.