From aeacfca5aea5df7c107cf0c12e72ab5d496c96e1 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Tue, 3 Jan 2023 19:48:17 +0100
Subject: More improvements to knowledge base

---
 source/know/concept/ghz-paradox/index.md | 41 +++++++++++++++++---------------
 1 file changed, 22 insertions(+), 19 deletions(-)

(limited to 'source/know/concept/ghz-paradox')

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
-- 
cgit v1.2.3