Exercise 3.4 Discussion

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf {Exercise\ 3.4}$

Show that the state

\begin{align} \ket{GHZ_n} = \frac{1}{\sqrt{2}} (\ket{00\dots 0}+ \ket{11\dots 1}) \end{align}

is entangled, with respect to the decompostion into the $n$ qubits, for every $n>1$.

page revision: 0, last edited: 11 Dec 2011 23:12

Assume $\ket{GHZ_n}$ is not entangled and $n > 1$. Then $\ket{GHZ_n}$ can be written as

(1)From the definition of $\ket{GHZ_n}$, we must therefore have

(2)so all of the $a_i$ values must be non-zero; and

(3)so all of the $b_i$ values must be non-zero. But we also have

(4)which is clearly impossible. Thus our assumption was wrong and $\ket{GHZ_n}$ must be entangled.

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