Exercise 2.6 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.6}$

For each pair consisting of a state and a measurement basis, describe the possible measurement outcomes and give the probability for each outcome.

a) $\frac{\sqrt 3}{2}\ket 0 - \frac{1}{2}\ket 1$, $\{\ket 0, \ket 1\}$

b) $\frac{\sqrt 3}{2}\ket 1 -\frac{1}{2}\ket 0$, $\{\ket 0, \ket 1\}$

c) $\ket{-\i}$, $\{\ket 0, \ket 1\}$

d) $\ket{0}$, $\{\ket{+}, \ket{-}\}$

e) $\frac{1}{\sqrt 2}\left(\ket{0} - \ket 1 \right)$, $\{\ket{\i}, \ket{-\i}\}$

f) $\ket{1}$, $\{\ket{\i}, \ket{-\i}\}$

g) $\ket{+}$, $\{\frac{1}{2}\ket{0} + \frac{\sqrt 3}{2}\ket 1, \frac{\sqrt 3}{2}\ket{0} - \frac{1}{2}\ket 1 \}$