3.2 Solution with Bell States
twhittle 21 Dec 2013 02:38
Consider the linear combination of the entangled Bell states below:
(1)\begin{align} a|\phi^+\rangle+b|\phi^-\rangle \end{align}
If we then choose $a=1/\sqrt2$ and $b=1/\sqrt2$, then the state above can be simplified:
(2)\begin{align} \frac{1}{\sqrt2}|\phi^+\rangle+\frac{1}{\sqrt2}|\phi^-\rangle = \frac{1}{2}(|00\rangle+|11\rangle+|00\rangle-|11\rangle)=|00\rangle \end{align}
Now we must find an a1, a2, b1, b2 that satisfy:
(3)\begin{align} a_1|0\rangle+b_1|1\rangle\otimes a_2|0\rangle+b_2|1\rangle=|00\rangle \end{align}
If we choose $a_1=1$, $b_1=0$, $a_2=1$ $b_2=0$, then we have satisfied the above relation. Thus we have shown that the linear combination of an entangled state is not necessarily entangled.