Categories: Physics, Quantum information, Quantum mechanics.

The **Greenberger-Horne-Zeilinger** or **GHZ paradox** is an alternative proof of Bell’s 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.

- N. Brunner,
*Quantum information theory: lecture notes*, 2019, unpublished. - J.B. Brask,
*Quantum information: lecture notes*, 2021, unpublished.

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