The Greenberger-Horne-Zeilinger or GHZ paradox
is an alternative proof of Bell’s theorem
that does not use inequalities, but instead
the three-particle entangled GHZ state :
Where and are qubit states,
specifically the eigenvalues of the Pauli matrix .
If we now apply certain products of the Pauli matrices and
as quantum gates
to this three-particle state, we find:
In other words, the GHZ state is a simultaneous eigenstate of these composite operators,
with eigenvalues and , respectively.
Let us do the same for two more operators,
so that we have a set of four observables
for which gives these eigenvalues:
According to any local hidden variable (LHV) theory,
the measurement outcomes of the operators are predetermined,
and the three particles , and can be measured separately,
or in other words, the eigenvalues can be factorized:
Let us now multiply both sides of these four equations together:
This is a contradiction: the left-hand side is ,
but all six factors on the right are .
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 could be measured on its own
without an “action-at-a-distance” effect on or .
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,
- J.B. Brask,
Quantum information: lecture notes,