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  • Open Access

Sparse anisotropic positive maps for qutrit entanglement: Exact indecomposability and positive-partial-transpose geometry

Massimiliano F. Sacchi*

  • *Contact author: massimiliano.sacchi@cnr.it
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Phys. Rev. A 114, 042455 – Published 26 October, 2026

DOI: https://doi.org/10.1103/lbj6-bts5

Abstract

Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters w and z. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square 0≤w,z≤2/3, complete positivity holds on the smaller square 0≤w,z≤1/3, and decomposability is lost precisely outside a quarter circle in the corner w,z≥1/3. The indecomposable region is certified by explicit PPT entangled states adapted to the same witness geometry. At the end point W*=W(2/3,2/3) we construct a four-parameter family of PPT edge states of rank type (5,5), derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although W* is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.

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Phys. Rev. A 114, 042455– Published 26 October, 2026
  • Received 19 June 2026
  • Accepted 14 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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