Competing coherence geometries in positive maps: positivity hierarchies, atomicity, and optimal entanglement witnesses

We introduce a family of positive maps on qudits whose Choi operators combine two competing coherence patterns: independent exchange couplings and a collective set of diagonal couplings. For every mixed member of the family, we derive exact conditions for positivity, complete positivity, and all intermediate levels of k-positivity, and we determine the complete phase structure of decomposability and atomicity. The higher-positivity thresholds depend on the channel multiplicities, whereas the exact decomposability boundary is universal in scale-free coordinates. Its proof combines an explicit positive-plus-copositive decomposition with a matching family of positive-partial-transpose (PPT) states. The latter form PPT edge states whose detection region is an ellipse, or a disk after collective rescaling. An embedded-qutrit certificate shows that indecomposability and atomicity coincide throughout the mixed family. In contrast, the purely collective endpoint is decomposable over its entire positive interval. Finally, at maximal coherence, an explicit positive subtraction produces an atomic optimal entanglement witness with the spanning property and a strictly larger PPT- detection set. The study thus provides an exactly solvable setting in which k-positivity, decomposability, atomicity, PPT-edge-state geometry, and witness optimality can be compared in arbitrary local dimension.

Source: arXiv:2610.35758v1 - http://arxiv.org/abs/2610.35758v1 PDF: https://arxiv.org/pdf/2610.35758v1 Original Link: http://arxiv.org/abs/2610.35758v1