Can Quantum Markov Evolutions Ever Be Dynamically Decoupled?

We consider the class of quantum stochastic evolutions ($SLH$-models) leading to a quantum dynamical semigroup over a fixed quantum mechanical system (taken to be finite-dimensional). We show that if the semigroup is dissipative, that is, the coupling operators are non-zero, then a dynamical decoupling scheme based on unitary rotations on the system space cannot suppress decoherence even in the limit where the period between pulses vanishes. We emphasize the role of the Fock space dilation used here to construct a quantum stochastic model, as there are often dilations of the same semigroup using an environmental noise model of lower level of chaoticity for which dynamical decoupling is effective. We show that the Chebotarev-Gregoratti Hamiltonian behind a quantum stochastic evolution is an example of a Hamiltonian dynamics on a joint system-environment that cannot be dynamically decoupled in this way.