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We apply the classical data processing inequality to quantum metrology toshow that manipulating the classical information from a quantum measurementcannot aid in the estimation of parameters encoded in quantum states. Wefurther derive a quantum data processing inequality to show that coherentmanipulation of quantum data also cannot improve the precision in estimation.In addition, we comment on the assumptions necessary to arrive at theseinequalities and how they might be avoided providing insights into enhancementprocedures which are not provably wrong.

The best Fisher is upstream: data processing inequalities for quantum metrology