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+---
+title: "GHZ paradox"
+date: 2021-03-29
+categories:
+- Physics
+- Quantum mechanics
+- Quantum information
+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,
+
+$$\begin{aligned}
+    \boxed{
+        \Ket{\mathrm{GHZ}}
+        = \frac{1}{\sqrt{2}} \Big( \Ket{000} + \Ket{111} \Big)
+    }
+\end{aligned}$$
+
+Where $\Ket{0}$ and $\Ket{1}$ are qubit states,
+for example, the eigenvalues of the Pauli matrix $\hat{\sigma}_z$.
+
+If we now apply certain products of the Pauli matrices $\hat{\sigma}_x$ and $\hat{\sigma}_y$
+to the three particles, we find:
+
+
+$$\begin{aligned}
+    \hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x \Ket{\mathrm{GHZ}}
+    &= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \Ket{0} \otimes \hat{\sigma}_x \Ket{0} \otimes \hat{\sigma}_x \Ket{0}
+    + \hat{\sigma}_x \Ket{1} \otimes \hat{\sigma}_x \Ket{1} \otimes \hat{\sigma}_x \Ket{1} \Big)
+    \\
+    &= \frac{1}{\sqrt{2}} \Big( \Ket{1} \otimes \Ket{1} \otimes \Ket{1} + \Ket{0} \otimes \Ket{0} \otimes \Ket{0} \Big)
+    = \Ket{\mathrm{GHZ}}
+    \\
+    \hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y \Ket{\mathrm{GHZ}}
+    &= \frac{1}{\sqrt{2}} \Big( \hat{\sigma}_x \Ket{0} \otimes \hat{\sigma}_y \Ket{0} \otimes \hat{\sigma}_y \Ket{0}
+    + \hat{\sigma}_x \Ket{1} \otimes \hat{\sigma}_y \Ket{1} \otimes \hat{\sigma}_y \Ket{1} \Big)
+    \\
+    &= \frac{1}{\sqrt{2}} \Big( \Ket{1} \otimes i \Ket{1} \otimes i \Ket{1} + \Ket{0} \otimes i \Ket{0} \otimes i \Ket{0} \Big)
+    = - \Ket{\mathrm{GHZ}}
+\end{aligned}$$
+
+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:
+
+$$\begin{aligned}
+    \hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x
+    \quad &\implies \quad +1
+    \\
+    \hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y
+    \quad &\implies \quad -1
+    \\
+    \hat{\sigma}_y \otimes \hat{\sigma}_x \otimes \hat{\sigma}_y
+    \quad &\implies \quad -1
+    \\
+    \hat{\sigma}_y \otimes \hat{\sigma}_y \otimes \hat{\sigma}_x
+    \quad &\implies \quad -1
+\end{aligned}$$
+
+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,
+or in other words, the eigenvalues can be factorized:
+
+$$\begin{aligned}
+    \hat{\sigma}_x \otimes \hat{\sigma}_x \otimes \hat{\sigma}_x
+    \quad &\implies \quad +1 = m_x^A m_x^B m_x^C
+    \\
+    \hat{\sigma}_x \otimes \hat{\sigma}_y \otimes \hat{\sigma}_y
+    \quad &\implies \quad -1 = m_x^A m_y^B m_y^C
+    \\
+    \hat{\sigma}_y \otimes \hat{\sigma}_x \otimes \hat{\sigma}_y
+    \quad &\implies \quad -1 = m_y^A m_x^B m_y^C
+    \\
+    \hat{\sigma}_y \otimes \hat{\sigma}_y \otimes \hat{\sigma}_x
+    \quad &\implies \quad -1 = m_y^A m_y^B m_x^C
+\end{aligned}$$
+
+Where $m_x^A = \pm 1$ etc.
+Let us now multiply both sides of these four equations together:
+
+$$\begin{aligned}
+    (+1) (-1) (-1) (-1)
+    &= (m_x^A m_x^B m_x^C) (m_x^A m_y^B m_y^C) (m_y^A m_x^B m_y^C) (m_y^A m_y^B m_x^C)
+    \\
+    -1
+    &= (m_x^A)^2 (m_x^B)^2 (m_x^C)^2 (m_y^A)^2 (m_y^B)^2 (m_y^C)^2
+\end{aligned}$$
+
+This is a contradiction: the left-hand side is $-1$,
+but all six factors on the right are $+1$.
+This 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 $C$.
+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.
+
+
+
+## References
+1.  N. Brunner,
+    *Quantum information theory: lecture notes*,
+    2019, unpublished.
+2.  J.B. Brask,
+    *Quantum information: lecture notes*,
+    2021, unpublished.
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