Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay
Abstract
We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the 2⊗3 system, yields, for every mn≥6, an explicit upper bound on the purity of APPT states that is asymptotic to 4/(3mn). This improves the previous 2/(mn) upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every 2⊗ n system with n≥3. For every 3⊗ n system with n≥3, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every 2⊗ n system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume an whose ratio is less than $4$ and tends to $3$. Consequently, an=Θ (√n(4/27)n), and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate ln(27/4).
Related
- Purity maxima, absolute separability, and spectral volumes of absolutely PPT states
- Maximum purity of absolutely separable and absolutely PPT states in qutrit--qudit systems
- Optimization of capital injections and absolutely continuous dividend payments in a diffusion model
- Relative entropy representations via tracial joint spectral measures
- Variance of relative entropy over random states
BibTeX
@misc{tran2026spectral,
title = {{Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay}},
author = {Anh T. Tran},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18568},
eprint = {2609.18568},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18568},
}
RIS
TY - UNPB
TI - Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay
AU - Anh T. Tran
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18568
UR - https://arxiv.org/abs/2609.18568
AB - We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the 2⊗3 system, yields, for every mn≥6, an explicit upper bound on the purity of APPT states that is asymptotic to 4/(3mn). This improves the previous 2/(mn) upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every 2⊗ n system with n≥3. For every 3⊗ n system with n≥3, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every 2⊗ n system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume a_n whose ratio is less than $4$ and tends to $3$. Consequently, a_n=Θ (√n(4/27)^n), and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate ln(27/4).
ER -