$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 4.7}$

Using the projection operator formalism

a) compute the probability of each of the possible outcomes of measuring the first qubit of an arbitrary two-qubit state in the Hadamard basis $\{\ket +, \ket - \}$.

b) compute the probability of each outcome for such a measurement on the state $\ket{\Psi^+} = \frac{1}{\sqrt 2}(\ket{00} + \ket{11}$.

c) for each possible outcome in part b), describe the possible outcomes if we now measure the second qubit in the standard basis.

d) for each possible outcome in part b), describe the possible outcomes if we now measure the second qubit in the Hadamard basis.