## Nielsen and Chuang Exercise 2.58

Exercise 2.58Supposed we prepare a quantum system in an eigenstate of some observable , with corresponding eigenvalue . What is the average observed value of , and the standard deviation.

As long as the state does not decohere, the average value will always be exactly , and the standard deviation will be 0. This does not violate the uncertainty principle, which is about pairs of noncommuting observables.

## Nielsen and Chuang Exercise 2.57

Exercise 2.57Cascaded measurements are single measurements. Suppose and are two sets of measurement operators. Show that a measurement defined by the measurement operators followed by a measurement defined by the measurement operators is physically equivalent to a single measurement defined by the measurement operators with the representation .

Suppose we consider the matrix element of the specified cascaded measurement. We can insert a complete set of states between the two operators via , making this equivalent to the matrix multiplication of elements of .

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