In this paper, near the bottom of the left half of page 3, the authors claim that any local Hermitian operator (one which acts only on a single subsystem of a larger composite system) can be expressed as a sum over local error operators. This is not immediately obvious to me but seems to be obvious to the authors. Can someone please explain why this is the case?
The authors define error operators as follows: Give a code on a composite quantum system, let $P$ be the projection onto the corresponding code space. A local error operator is defined as any operator $E$ such that $PEP \propto P$.