Extend the decomposition theorem to deficient-Choi-rank channels

Extend the quantitative decomposition theorem for overall T-violation, nonunitality, and kinematic T-violation to quantum channels whose Choi states have deficient Choi rank.

Background

The paper establishes a relative-entropy Pythagorean decomposition of overall T-violation into nonunitality and the residual kinematic T-violation of a closest unital channel. The proof is stated for channels with strictly positive Choi states, and an additional e-type identity is obtained for two-dimensional systems.

Channels with deficient Choi rank fall outside the strict-positivity condition used in the theorem. The authors explicitly state that they do not currently have an extension of the decomposition theorem to this case, leaving open whether an analogous identity can be established for such channels.