Weak values with decoherence

The weak value of an observable is experimentally accessible by weak measurements as theoretically analyzed by Aharonov et al and recently experimentally demonstrated. We introduce a W operator associated with the weak values and give a general framework of quantum operations to the W operator in parallel with the Kraus representation of the completely positive map for the density operator. The decoherence effect is also investigated in terms of the weak measurement by a shift of a probe wavefunction of continuous variable. As an application, we demonstrate how the geometric phase is affected by the bit flip noise.