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. By a version of the Heisenberg Principle, which we prove, such a protected space must be “dark” in the sense that no information leaks out during the transmission. We explain the role of the Knill–Laflamme condition in relation to protection and darkness, and we analyze several degrees of protection, whether related to error correction, or to state restauration after a measurement. 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 tri-unitary noise models.