The geometry of absolute separability and other convex matrix properties from spectrum
Abstract
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), \emph{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. While ASEP2,n is a spectrahedron, too, for ASEPm,n we can in general only show that it is a semialgebraic set for all m,n≥ 3. 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.
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BibTeX
@misc{ahiable2026geometry,
title = {{The geometry of absolute separability and other convex matrix properties from spectrum}},
author = {Jennifer Ahiable and Naga Bhavya Teja Kothakonda and Andreas Winter},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03390},
eprint = {2608.03390},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03390},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The geometry of absolute separability and other convex matrix properties from spectrum
AU - Jennifer Ahiable
AU - Naga Bhavya Teja Kothakonda
AU - Andreas Winter
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.03390
UR - https://arxiv.org/abs/2608.03390
AB - 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), \emph{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≤ 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≥ 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.
ER -