Uncertainty relations for generalized quantum measurements and completely positive maps

The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise ε brings the momentum disturbance not less than h/2ε. However, this relation holds only for restricted class of measurements. Here, a generalized uncertainty relation for measurement noise and disturbance is formalized and proven, which holds for all the possible quantum measurements. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. The measurement noise and disturbance are defined rigorously for any such general quantum measurements.