summaryrefslogtreecommitdiff
path: root/source/know/concept/ghz-paradox
diff options
context:
space:
mode:
Diffstat (limited to 'source/know/concept/ghz-paradox')
-rw-r--r--source/know/concept/ghz-paradox/index.md41
1 files changed, 22 insertions, 19 deletions
diff --git a/source/know/concept/ghz-paradox/index.md b/source/know/concept/ghz-paradox/index.md
index a59ccfe..9951883 100644
--- a/source/know/concept/ghz-paradox/index.md
+++ b/source/know/concept/ghz-paradox/index.md
@@ -12,43 +12,46 @@ layout: "concept"
The **Greenberger-Horne-Zeilinger** or **GHZ paradox**
is an alternative proof of [Bell's theorem](/know/concept/bells-theorem/)
that does not use inequalities,
-but the three-particle entangled **GHZ state** $$\Ket{\mathrm{GHZ}}$$ instead,
+but the three-particle entangled **GHZ state** $$\ket{\mathrm{GHZ}}$$ instead,
$$\begin{aligned}
\boxed{
- \Ket{\mathrm{GHZ}}
- = \frac{1}{\sqrt{2}} \Big( \Ket{000} + \Ket{111} \Big)
+ \ket{\mathrm{GHZ}}
+ = \frac{1}{\sqrt{2}} \Big( \ket{000} + \ket{111} \Big)
}
\end{aligned}$$
-Where $$\Ket{0}$$ and $$\Ket{1}$$ are qubit states,
-for example, the eigenvalues of the Pauli matrix $$\hat{\sigma}_z$$.
+Where $$\ket{0}$$ and $$\ket{1}$$ are qubit states,
+specifically the eigenvalues of the Pauli matrix $$\hat{\sigma}_z$$.
If we now apply certain products of the Pauli matrices $$\hat{\sigma}_x$$ and $$\hat{\sigma}_y$$
-to the three particles, we find:
+as [quantum gates](/know/concept/quantum-gate/)
+to this three-particle state, we find:
$$\begin{aligned}
- \hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x \Ket{\mathrm{GHZ}}
- &= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \Ket{0} \otimes \hat{\sigma}_x \Ket{0} \otimes \hat{\sigma}_x \Ket{0}
- + \hat{\sigma}_x \Ket{1} \otimes \hat{\sigma}_x \Ket{1} \otimes \hat{\sigma}_x \Ket{1} \Big)
+ \hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x \ket{\mathrm{GHZ}}
+ &= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \ket{0} \otimes \hat{\sigma}_x \ket{0} \otimes \hat{\sigma}_x \ket{0}
+ + \hat{\sigma}_x \ket{1} \otimes \hat{\sigma}_x \ket{1} \otimes \hat{\sigma}_x \ket{1} \Big)
\\
- &= \frac{1}{\sqrt{2}} \Big( \Ket{1} \otimes \Ket{1} \otimes \Ket{1} + \Ket{0} \otimes \Ket{0} \otimes \Ket{0} \Big)
- = \Ket{\mathrm{GHZ}}
+ &= \frac{1}{\sqrt{2}} \Big( \ket{1} \otimes \ket{1} \otimes \ket{1} + \ket{0} \otimes \ket{0} \otimes \ket{0} \Big)
\\
- \hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y \Ket{\mathrm{GHZ}}
- &= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \Ket{0} \otimes \hat{\sigma}_y \Ket{0} \otimes \hat{\sigma}_y \Ket{0}
- + \hat{\sigma}_x \Ket{1} \otimes \hat{\sigma}_y \Ket{1} \otimes \hat{\sigma}_y \Ket{1} \Big)
+ &= \ket{\mathrm{GHZ}}
\\
- &= \frac{1}{\sqrt{2}} \Big( \Ket{1} \otimes i \Ket{1} \otimes i \Ket{1} + \Ket{0} \otimes i \Ket{0} \otimes i \Ket{0} \Big)
- = - \Ket{\mathrm{GHZ}}
+ \hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y \ket{\mathrm{GHZ}}
+ &= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \ket{0} \otimes \hat{\sigma}_y \ket{0} \otimes \hat{\sigma}_y \ket{0}
+ + \hat{\sigma}_x \ket{1} \otimes \hat{\sigma}_y \ket{1} \otimes \hat{\sigma}_y \ket{1} \Big)
+ \\
+ &= \frac{1}{\sqrt{2}} \Big( \ket{1} \otimes i \ket{1} \otimes i \ket{1} + \ket{0} \otimes i \ket{0} \otimes i \ket{0} \Big)
+ \\
+ &= - \ket{\mathrm{GHZ}}
\end{aligned}$$
In other words, the GHZ state is a simultaneous eigenstate of these composite operators,
with eigenvalues $$+1$$ and $$-1$$, respectively.
-Let us introduce two other product operators,
-such that we have a set of four observables,
-for which $$\Ket{\mathrm{GHZ}}$$ gives these eigenvalues:
+Let us introduce two more operators in the same way,
+so that we have a set of four observables,
+for which $$\ket{\mathrm{GHZ}}$$ gives these eigenvalues:
$$\begin{aligned}
\hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x