## Nielsen and Chuang Exercise 2.66

Exercise 2.66Show that the average value of the observable for a two-qubit system measured in the state is zero.

Twice the average value is given by

## Nielsen and Chuang Exercise 2.65

Exercise 2.65Express the states and in a basis in which they arenotthe same up to a relative phase shift.

This is perhaps a trivial transformation, but, if we express the two states in the eigenbasis of the operator, the states and correspond to the and states, which no longer differ by merely a relative phase shift.

## Nielsen and Chuang Exercise 2.64

Exercise 2.64Suppose Bob is given a quantum state chosen from a set of linearly independent states. Construct a POVM such that if outcome occurs, , then Bob knows with certainty that he was given the state . The POVM must be such that for each .

Since the states are linearly independent, we can define , for , and , which satisfies , and insures if is measured, that the system is in state .

## Nielsen and Chuang Exercise 2.63

Exercise 2.63Suppose a measurement is described by measurement operators . Show that there exist unitary operators such that , where is the POVM associated to the measurement.

We can confirm that the proposed operator works by considering that . Thus, if , which is the case if the represent a POVM, then , which demonstrates that the unitary transformation converts the measurement operators to the POVMs.

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