Classification of Kirkwood-Dirac positivity-preserving unitaries
The paper classifies unitary dynamics preserving Kirkwood-Dirac (KD) positivity into three types: I, II, and III gates. It shows that for specific dimensions ($d=pq$), type III gates cause exponential simulation overhead when inputs are non-real. This discovery proves a fundamental obstruction for resource theories built on these distributions, meaning they cannot treat total non-positivity as a monotone while allowing all efficiently simulable positivity-preserving unitaries to be free operations.