Extended Controllability Tests for Quantum Decoherence-Free Subspaces

In this paper we develop two axiomatic tests for the controllability of subsystem codes embedded in decoherence-free subspaces of open quantum systems. The tests expand on existing control theory by considering quantum subsystems where a decoherence-protected quantum state is permitted to exit the set of logically encoded states in order to perform a broader range of computations. The tests target the class of all Lindbladian models and require no specific structure, regularity, or symmetry in system Hamiltonians or noise operators, making them ideal for control design in models lacking these features. The usefulness of the tests is demonstrated using a complete worked example for a trapped ion system subject to a nonstandard collective dephasing noise.