Exercise 2.2

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.2}$

Which pairs of expressions for quantum states represent the same state? For those pairs that represent different states, describe a measurement for which the probabilities of the two outcomes differ for the two states and give these probabilities.

a) $\ket 0$ and $-\ket 0$

b) $\ket 1$ and $\i\ket 1$

c) $\frac{1}{\sqrt 2}\left(\ket 0 + \ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(-\ket 0 + \i\ket 1 \right)$

d) $\frac{1}{\sqrt 2}\left(\ket 0 + \ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(\ket 0 - \ket 1 \right)$

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

f) $\frac{1}{\sqrt 2}\left(\ket 0 + \i\ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(\i\ket 1 - \ket 0 \right)$

g) $\frac{1}{\sqrt 2}\left(\ket{+} + \ket{-} \right)$ and $\ket 0$

h) $\frac{1}{\sqrt 2}\left(\ket{\i} - \ket{-\i} \right)$ and $\ket 1$

i) $\frac{1}{\sqrt 2}\left(\ket{\i} + \ket{-\i} \right)$ and $\frac{1}{\sqrt 2}\left(\ket{-} + \ket{+} \right)$

j) $\frac{1}{\sqrt 2}\left(\ket{0} + e^{\i\pi/4}\ket{1} \right)$ and $\frac{1}{\sqrt 2}\left(e^{-\i\pi/4}\ket{0} + \ket{1} \right)$