Non-uniqueness of local description for all maps in SN \ SP

Establish that for every Hermitian-preserving trace-preserving map that is semi-nonnegative but not semi-positive, the local system evolution cannot be uniquely characterized by the map itself, i.e., distinct physical processes can yield the same effective local transformation.

Background

In the supplemental material, the authors present a single-qubit example of a semi-nonnegative but not semi-positive map \Psi for which the local process in a circuit segment is not uniquely determined by \Psi—any unitary that preserves a specific pure state can replace \Psi in the construction, leading to the same effective behavior.

They generalize this observation by conjecturing that such non-uniqueness holds for all maps in the set \mathcal{SN} \backslash \mathcal{SP}, which captures semi-nonnegative maps that are not semi-positive.