Exercise 3.7 Discussion
$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 3.7}$
Write the following states in terms of the Bell basis.
a) $\ket{00}$
b) $\ket{+}\ket{-}$
c) $\frac{1}{\sqrt 3}(\ket{00} + \ket{01} + \ket{10})$
page revision: 1, last edited: 11 Dec 2011 23:16
Using the notation for Bell states given in the text:
a. $\ket{00} = \frac{1}{\sqrt{2}} \big( \ket{\Phi^{+}} + \ket{\Phi^{-}} \big)$,
b. $\ket{+}\ket{-} = \frac{1}{2}\big( \ket{00} - \ket{01} + \ket{10} - \ket{11}\big) = \frac{1}{\sqrt{2}}\big( \ket{\Phi^{-}} - \ket{\Psi^{-}}\big)$, and
c. $\frac{1}{\sqrt{3}}\big( \ket{00} + \ket{01} + \ket{10}\big) = \frac{1}{\sqrt{6}}\ket{\Phi^{+}} + \frac{1}{\sqrt{6}}\ket{\Phi^{-}} + \sqrt{\frac{2}{3}} \ket{\Psi^{+}}$.
IMHO, a better presentation of c) would be
$\frac{1}{\sqrt{6}} (\ket{\Phi^+} + \ket{\Phi^-} + 2\ket{\Psi^+} )$
which reveals a nice structure. (Sorry for nitpicking).
How would one come to the solution for c? It doesn't seem obvious to me.
$\ket{00}$ comes from point a. $\ket{01}+\ket{10}$ is the defintion of $\ket{\Psi^+}$ (up to a coefficient).