Derivation of Ozawa’s Uncertainty Relation

The starting point in the derivation of Ozawa’s generalized error-disturbance uncertainty relation is the fact that , since they represent operators from different Hilbert spaces. Next we apply a unitary time-evolution of the composite quantum system, consisting of object system and measurement device  . Using the definition for noise and disturbance operators and we get . Taking the modulus and applying triangular inequality yields , where , with and being the initial state of object and measurement device system, respectively. With the definition of error and standard deviation we get . The same applies to the disturbance , which gives . Now we can apply Robertson relation three times: , and , which all together results in Ozawa’s generalized error-disturbance uncertainty relation .

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