# Local Hermitian operators can be written as sums over local error operators?

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 $$PEP \propto P$$.

• what is local error operator? Mar 29 at 12:38
• I've added the definition to the post. Mar 29 at 12:51
• isn't it just the definition of local error-detecting codes? For example, if the underlying system is a bunch of qubits, you can express any Hermitian operator as a sum over Paulis (because Paulis form a basis). If it is also local it means each term of the sum has few (constant) number of Paulis which is less than the code distance. So each term in the sum is a detectable error, i.e. local error operator (see the first equation in page 1). Apr 20 at 4:13