Maximum purity of absolutely separable and absolutely PPT states in qutrit--qudit systems

@misc{tran2026maximum,
  title = {{Maximum purity of absolutely separable and absolutely PPT states in qutrit--qudit systems}},
  author = {Anh T. Tran},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01306},
  eprint = {2611.01306},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01306},
}

TY  - UNPB
TI  - Maximum purity of absolutely separable and absolutely PPT states in qutrit--qudit systems
AU  - Anh T. Tran
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01306
UR  - https://arxiv.org/abs/2611.01306
AB  - A bipartite state is absolutely positive partial transpose (APPT) if it remains PPT under every global unitary transformation, and absolutely separable if it remains separable. For every qutrit--qudit system ℂ^3⊗ℂ^n, n≥3, we prove that the maximum purity is the same for absolutely separable and APPT states and equals max{(3n+8)/(3n+2)^2, 3/(8n)}. The maximizing ordered spectrum is unique and contains exactly two distinct eigenvalues: the larger one has multiplicity one for 3≤ n≤8 and multiplicity n for n≥9. We also derive, in terms of the purity, an upper bound on the minimum partial-transpose eigenvalue achievable by global unitary transformations, and characterize all states for which the bound is attained. These results confirm the inscribed-polytope purity conjecture of Ahiable, Kothakonda, and Winter for all qutrit--qudit systems. They also give a lower bound, sharp over explicit purity ranges, on the global depolarizing noise needed for absolute separability or APPT.
ER  -