Toroidal-Core Certificates for PPT-Squared Diagonal Orthogonal Covariant Channels
Abstract
The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps ΦA,B,C and ΦD,E,F on Md, the entrywise bound (AD)ij≥ 2⌊√d⌋√(AiiAjjDiiDjj) for all i,j guarantees that ΦA,B,C∘ΦD,E,F is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with a≠1 is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.
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BibTeX
@misc{gonzalez2026toroidalcore,
title = {{Toroidal-Core Certificates for PPT-Squared Diagonal Orthogonal Covariant Channels}},
author = {Samuel Marquez Gonzalez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32979},
eprint = {2609.32979},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32979},
}
RIS
TY - UNPB
TI - Toroidal-Core Certificates for PPT-Squared Diagonal Orthogonal Covariant Channels
AU - Samuel Marquez Gonzalez
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32979
UR - https://arxiv.org/abs/2609.32979
AB - The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps Φ_A,B,C and Φ_D,E,F on M_d, the entrywise bound (AD)_ij≥ 2⌊√d⌋√(A_iiA_jjD_iiD_jj) for all i,j guarantees that Φ_A,B,C∘Φ_D,E,F is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with a≠1 is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.
ER -