4. (20 points total) The Hadamard basis for one qubit is defined as: |0) + |1)...

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4. (20 points total) The Hadamard basis for one qubit is defined as: |0) + |1) (0) - 11) √2 √2 with corresponding projectors P+ = +)(+| and P_ := |-)(-|. |+) :: = 9 1 1-) == (a) (5 points) Write P+ and P- as 2 × 2 matrices with respect to the standard (computa- tional) basis (10), [1)}. (b) (5 points) Using your answer to (a), write P+ P_ as a 4 × 4 matrix. (c) (10 points) Suppose the state [0) is projectively measured with respect to the Hadamard basis. What is the probability of the outcome "+"? 4. (20 points total) The Hadamard basis for one qubit is defined as: |0) + |1) (0) - 11) √2 √2 with corresponding projectors P+ = +)(+| and P_ := |-)(-|. |+) :: = 9 1 1-) == (a) (5 points) Write P+ and P- as 2 × 2 matrices with respect to the standard (computa- tional) basis (10), [1)}. (b) (5 points) Using your answer to (a), write P+ P_ as a 4 × 4 matrix. (c) (10 points) Suppose the state [0) is projectively measured with respect to the Hadamard basis. What is the probability of the outcome "+"?