Exercise 2.4 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.4}$

Which of the states in 2.3 are superpositions with respect to the Hadamard basis, and which are not?

States from exercise 2.3:

a) $\ket +$

b) $\frac{1}{\sqrt 2}(\ket + + \ket -)$

c) $\frac{1}{\sqrt 2}(\ket + - \ket -)$

d) $\frac{\sqrt 3}{2}\ket + - \frac{1}{2} \ket -)$

e) $\frac{1}{\sqrt 2}(\ket\i - \ket{-\i} )$

f) $\frac{1}{\sqrt 2}(\ket 0 - \ket 1)$

page revision: 1, last edited: 12 Nov 2011 05:42

The states in (b), (c), and (d) are clearly superpositions with respect to the Hadamard basis. The state (e) is also a superposition since

(1)This leaves the states in (a) and (f), which are not superpositions with respect to the Hadamard basis. The state in (a) is $\ket{+}$ and the state in (f) is equal to $\ket{-}$, which are the Hadamard basis vectors.

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