From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/ghz-paradox/index.md | 24 ++++++++++++------------
 1 file changed, 12 insertions(+), 12 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 e1129d4..a59ccfe 100644
--- a/source/know/concept/ghz-paradox/index.md
+++ b/source/know/concept/ghz-paradox/index.md
@@ -12,7 +12,7 @@ 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,
 
 $$\begin{aligned}
     \boxed{
@@ -21,10 +21,10 @@ $$\begin{aligned}
     }
 \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,
+for example, the eigenvalues of the Pauli matrix $$\hat{\sigma}_z$$.
 
-If we now apply certain products of the Pauli matrices $\hat{\sigma}_x$ and $\hat{\sigma}_y$
+If we now apply certain products of the Pauli matrices $$\hat{\sigma}_x$$ and $$\hat{\sigma}_y$$
 to the three particles, we find:
 
 
@@ -45,10 +45,10 @@ $$\begin{aligned}
 \end{aligned}$$
 
 In other words, the GHZ state is a simultaneous eigenstate of these composite operators,
-with eigenvalues $+1$ and $-1$, respectively.
+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:
+for which $$\Ket{\mathrm{GHZ}}$$ gives these eigenvalues:
 
 $$\begin{aligned}
     \hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x
@@ -66,7 +66,7 @@ $$\begin{aligned}
 
 According to any local hidden variable (LHV) theory,
 the measurement outcomes of the operators are predetermined,
-and the three particles $A$, $B$ and $C$ can be measured separately,
+and the three particles $$A$$, $$B$$ and $$C$$ can be measured separately,
 or in other words, the eigenvalues can be factorized:
 
 $$\begin{aligned}
@@ -83,7 +83,7 @@ $$\begin{aligned}
     \quad &\implies \quad -1 = m_y^A m_y^B m_x^C
 \end{aligned}$$
 
-Where $m_x^A = \pm 1$ etc.
+Where $$m_x^A = \pm 1$$ etc.
 Let us now multiply both sides of these four equations together:
 
 $$\begin{aligned}
@@ -94,13 +94,13 @@ $$\begin{aligned}
     &= (m_x^A)^2 (m_x^B)^2 (m_x^C)^2 (m_y^A)^2 (m_y^B)^2 (m_y^C)^2
 \end{aligned}$$
 
-This is a contradiction: the left-hand side is $-1$,
-but all six factors on the right are $+1$.
+This is a contradiction: the left-hand side is $$-1$$,
+but all six factors on the right are $$+1$$.
 This means that we must have made an incorrect assumption along the way.
 
 Our only assumption was that we could factorize the eigenvalues,
-so that e.g. particle $A$ could be measured on its own
-without an "action-at-a-distance" effect on $B$ or $C$.
+so that e.g. particle $$A$$ could be measured on its own
+without an "action-at-a-distance" effect on $$B$$ or $$C$$.
 However, because that leads us to a contradiction,
 we must conclude that action-at-a-distance exists,
 and that therefore all LHV-based theories are invalid.
-- 
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