  Sequences of commuting quantum operators can be parallelized using
entanglement. This transformation is behind some optimal quantum metrology
protocols and recent results on quantum circuit complexity. We show that
dephasing quantum maps in arbitrary dimension can also be parallelized. This
implies that for general dephasing noise the protocol with entanglement is not
more fragile than the corresponding sequential protocol and, conversely, the
sequential protocol is not less effective than the entangled one. We derive
this result using tensor networks. Furthermore, we only use transformations
strictly valid within string diagrams in dagger compact closed categories.
Therefore, they apply verbatim to other theories, such as geometric
quantization and topological quantum field theory. This clarifies and
characterizes to some extent the role of entanglement in general quantum
theories.
