Non-SP SNTP maps in higher dimensions: no information backflow and non-uniqueness

Prove that for higher-dimensional finite-dimensional Hilbert spaces, every Hermitian-preserving trace-preserving map that is semi-nonnegative but not semi-positive (that is, it maps at least one density matrix to a density matrix but does not map any invertible density matrix to an invertible density matrix) both signals no information backflow in the open-system dynamics and is not uniquely defined as a local system map.

Background

The paper introduces semi-nonnegative trace-preserving (SNTP) maps as Hermitian-preserving trace-preserving maps that send at least one density matrix to a density matrix. Semi-positive trace-preserving (SPTP) maps are those SNTP maps that additionally map some invertible density matrix to an invertible density matrix. The authors analyze physical interpretations of these maps beyond CPTP and relate SPTP maps to non-Markovian dynamics under CP-divisibility.

For single-qubit examples, non-SP SNTP maps tend to only map a single pure state to a pure state and otherwise output non-density matrices; in such scenarios the authors argue that information backflow does not occur and the effective local description is not unique. Extending this behavior to higher-dimensional Hilbert spaces is posed as a conjecture.