---
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.