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Diffstat (limited to 'source/know/concept/ghz-paradox/index.md')
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1 files changed, 12 insertions, 12 deletions
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. |