Geometry, symmetry, and topology of orientation‑reversing Pauli channels

Abstract

Pauli channels that reverse the orientation of the Bloch sphere are often regarded as incompatible with complete positivity because they are associated with the transpose map. Here we show that this belief is incorrect. Using the geometric representation of unital qubit channels, we identify four inversion domains inside the Pauli tetrahedron and describe their boundaries, how they touch each other, and their internal structure. Each domain contains nontrivial regions of completely positive and trace‑preserving maps, including neighborhoods near the center of the tetrahedron. This shows that orientation reversal is a robust interior feature of the CPTP set, rather than a boundary artifact. We also characterize the topology of these regions and show that the four inversion domains form a single Clifford‑equivalence class. This reveals a previously overlooked geometric and topological stratification of Pauli channels and clarifies the physical role of orientation‑reversing dynamics in the structure of unital qubit maps.