GHZ experiments

GHZ experiments provide evidence against local hidden-variable theories. They confirm the apparent nonlocal effects of QM. In this article, I'll explain a GHZ thought experiment using qubits. I aim to be as simple and concise as possible.

Notation

We'll work in the qubit basis \(\{\textbf{0},\textbf{1}\}\) and use the following other states in this article.

• \(\textbf{+}=\textbf{0}+\textbf{1}\)
• \(\textbf{-}=\textbf{0}-\textbf{1}\)
• \(\textbf{R}=\textbf{0}+i\textbf{1}\)
• \(\textbf{L}=\textbf{0}-i\textbf{1}\)
• \(\textbf{GHZ}=\textbf{000}+\textbf{111}\)

We won't worry about normalizing states and we simply bold symbols of quantum states (rather than using ket notation or other).

We will measure individual qubits either on \(\{\textbf{+},\textbf{-}\}\) or \(\{\textbf{R},\textbf{L}\}\). We refer to the first measurement as \(M^+_-\) and the latter as \(M^R_L\).

GHZ State

Notice the last state \(\textbf{GHZ}\) defined in the prior section. This is an entagled state of 3 qubits, similar to the more familiar singlet state \(\textbf{00}+\textbf{11}\). If any qubit is measured to be in state \(\textbf{0}\) we know the other 2 will have the same result if measured the same way. Likewise for \(\textbf{1}\).

Below we list a few states that are orthogonal to \(\textbf{GHZ}\), because they will be crucial to understanding this article. Recall that no measurement of \(\textbf{GHZ}\) can result in any of these perpendicular states. This is an experimentally verified prediction of QM.

• \(\textbf{+RR}\)
• \(\textbf{+LL}\)
• \(\textbf{-RL}\)
• \(\textbf{-LR}\)
• \(\textbf{++-}\)
• \(\textbf{+-+}\)

Experimental scenario

Suppose we create a GHZ state and measure \(M^+_-\) on the first qubit and \(M^R_L\) on the other 2. A key observation is that QM restricts the possible outcomes as follows.

• If the first measurement was \(\textbf{+}\), then exactly one of the last two qubits will be found in state \(\textbf{R}\) and the other will be found in state \(\textbf{L}\). This is because \(\textbf{+RR}\) and \(\textbf{+LL}\) are othrthogonal to \(\textbf{GHZ}\) (list above).
• If the first measurement was \(\textbf{-}\), then the last two qubits will have the same measurement result (both \(\textbf{R}\) or both \(\textbf{L}\)). This also follows from the fact that the other two combinations are orthogonal to \(\textbf{GHZ}\).
• There are four possible results, each of which occur with probability \(1/4\): \(\textbf{+RL},\textbf{+LR},\textbf{-RR},\textbf{-LL}\).

Dependency of measurement results

Notice that, if I tell you the results of any 2 measurements, you can perfectly predict the last one. For example, if the first 2 results are \(\textbf{+R}\) you know the last will be \(\textbf{L}\) (because only 1 of the 4 results starts with \(\textbf{+R}\)). This is true even if the last qubit is far from us, and measured outside the future light cone of the prior two measurements. This has been experimentally verified.

Pre-assignment doesn't work

At first, this may not seem to violate local hidden-variable theories. One could argue that the measurement results were "pre-assigned" long before the measurement ocurred, at a location/time that includes all the measurement events in its future light cone. For example, I created the qubits in my lab, they already "knew" they would provide the result \(\textbf{+RL}\), then I sent one qubit off to a distant galaxy, to be measured some time in the future. It turns out that, given seemingly natural assumptions, we can prove such an explanation wrong.

Let's explain why these results couldn't have been pre-assigned. A key assumption is that we have the freedom to choose what type of measurement we perform (\(M^+_-\) or \(M^R_L\)) after the third qubit moved away, outside the past light cone of its measurement. In other words, we are assuming the experimenters have free will to chose what experiments to perform.

Now we'll restrict our analysis to the cases where the qubits were supposedly pre-assigned \(\textbf{+RL}\). Let's say we instead performed \(M^R_L\) on the first qubit and \(M^+_-\) on the second. Since we know the last qubit is \(\textbf{L}\) there are two possible options for the first two: \(\textbf{R+}\) or \(\textbf{L-}\) (the other 2 combinations are orthogonal to \(\textbf{GHZ}\)).

Let's take the first option, assuming \(\textbf{R+L}\) has been preassigned (in addition to \(\textbf{+RL}\)). Now we know what must happen if we instead measure the last qubit on \(\{\textbf{+},\textbf{-}\}\). We know the first is \(\textbf{R}\) and the second is \(\textbf{R}\) and therefore the last must be \(\textbf{-}\) (because \(\textbf{RR+}\) is orthogonal to \(\textbf{GHZ}\)). By reviewing all the pre-assignments, we know that, if we perform \(M^+_-\) on all qubits, we'll get \(++-\). However, QM predicts, and experiments confirm, that \(++-\) can't be measured from \(GHZ\) (again, orthogonal), hence we have a contradiction to such pre-assignments.

Let's try to avoid this by assuming the second possible option \(\textbf{L-L}\) was pre-assigned. Again, if we perform \(M^+_-\) on the last qubit we know the result will be \(\textbf{+}\) because we know the first two are \(\textbf{LR}\) and \(\textbf{LR-}\) is orthogonal to \(\textbf{GHZ}\). Once again, though, if we collect all these pre-assignments we find that performing \(M^+_-\) on all qubits must result in \(\textbf{+-+}\) which is orthogonal to \(\textbf{GHZ}\) and hence impossible.

You can apply this logic too all the other possible combinations. There's simply no way to pre-assign results for all possible experimental choices. At some point you'll have to make a pre-assignment that's counter to both QM and experimental results.

Fair sampling loophole

In addition to the lack of free will, there's another weakness in our argument above against "preassignment." It stems from the fact that experiments fail to register results in some fraction of the trials. Perhaps we try to run the experiment 100 times, but we only capture 85 results.

We like to assume the 85 results we obtained are a "fair sample" from all the 100 experiments, meaning the statistics captured are essentially the same. However, this may not be the case. For example, nature could preassign results for each qubit, and in the rare case we choose to measure in a way that would reveal a contradiction to QM, nature causes the experimental trial to fail without a result. In this way nature might be able to preassign results, we just can't expose it via experimentations.

This loophole has been closed. Experimenters have been able to increase the "detection efficiency" (reduce the number of failed experiments) such that it would be impossible for nature to utilize the loophole and match QM predictions with preassignment.

Conclusion

QM predicts, and measurements confirm, that measurement results cannot be "pre-assigned" for all possible combinations of measurement choices. Measurement results apparently are "assigned" after knowing how the individual qubits will be measured.

We also saw that measurement results on one qubit can be perfectly determined by the results from measuring the other two. Assuming experimenters have free will to chose which measurement they perform, this implies each qubit must "choose" its measurement result as a function of the potentially space-like separated events of choosing and measuring the other two. This spooky results seems to defy local explanations including the fair sampling loophole.