Inverse Problem of Alchemical Resource Theory: Replication and Programming Single Out Imaginarity and Parity Asymmetry

Abstract

Quantum resource theories usually begin with a prescribed free structure and ask what tasks become possible when a resource is supplied. We study the inverse problem of alchemical resource theories, which we define as resource theories admitting a resource that can both replicate itself exactly and universally program quantum instruments. Under natural consistency assumptions and branchwise complete freeness, we show that for qubit single systems only two nontrivial theories survive, up to a common local unitary change of basis: parity asymmetry and imaginarity. We further show that exact universality exhibits an all-or-nothing trade-off: any nonmaximal resource state can exactly program only free unitaries. These results demonstrate that prescribed operational capabilities can strongly constrain the underlying resource structure and reveal a fundamental trade-off between computational capability and the precision required for physical implementation.