Protected Subspaces in Quantum Information

In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by 'coding' the state space of the system before transmission into a 'protected' part of a larger state space, and by applying a proper 'decoding' map afterwards. We investigate two kinds of protected subspaces: the error correction subspaces and the dark subspaces. In neither case information leaks out of the system, but in the second it is more difficult to restore the state. Recent results on higher rank numerical ranges of operators are used to construct examples. In particular, dark spaces are constructed for any map of rank 2, for a biased permutations channel and for certain separable maps acting on multipartite systems. Furthermore, error correction subspaces are provided for a class of three unitary noise models.