Quantum Operations for Weak Values

The weak value of an observable is experimentally accessible by weakmeasurements as Y. Aharonov et al. theoretically analyzed and recentlyexperimentally demonstrated. We give a general framework to describe effectsof quantum operation to the weak values in parallel with the Krausrepresentation of the completely positive map for the density operator. Asan application, we demonstrate how the geometric phase is affected by thebit flip noise since the geometric phase can be characterized by the weakvalues.