Noiseless Quantum Codes

We study a class of open quantum systems made by $N$ replicas of a given system S coupled with the environment in a replica independent way. If ${\cal A}_S\subset su(d)$ is the rank $r$ dynamical algebra of $S,$ the global Hilbert space splits according to the ${\cal A}_S$-irreps. For $N>r$ 1-dimensional irreps exist that are annihilated by the environment-system interaction. Out of these representations one can build subspaces over which the induced Liouvillian dynamics is unitary. These subspace can be thought of as noiseless quantum codes.