  The information flow in a quantum system is a fundamental feature of its
dynamics. An important class of dynamics are quantum cellular automata (QCA),
systems with discrete updates invariant in time and space, for which an index
theory has been proposed for the quantification of the net flow of quantum
information across a boundary. While the index is rigid in the sense of begin
invariant under finite-depth local circuits, it is not defined when the system
is coupled to an environment, i.e. for non-unitary time evolution of open
quantum systems. We propose a new measure of information flow for non-unitary
QCA denoted the information current which is not rigid, but can be computed
locally based on the matrix-product operator representation of the map.
