## Friday, December 30, 2011

### GHZM experiment and indefensible emulators of quantum mechanics

I want to go through the GHZM experiment again and somewhat carefully (and in $$\LaTeX$$) and discuss the insanity of the assumptions about the laws of Nature that are forced upon you if you want to believe in "realism", i.e. the idea that the results of experiments (including those at the microscopic level) reflect a pre-existing reality.

GHZM state

The Greenberger-Horne-Zeilinger state, later improved by Mermin, is an entangled state of three spins. Imagine that we produce 8 million triples of electrons and send the electrons to 3 laboratories – on Earth, Jupiter, and Saturn. The distance between each pair of these laboratories is 1 light hour.

These will be referred to as laboratories 1, 2, 3, and the corresponding electrons will be described by the basis vectors $$\ket\uparrow$$ and $$\ket\downarrow$$, meaning $$J_z=\pm \hbar/2$$, written at the three places of a tensor product.

Clearly, some familiarity with the bracket notation is required. Each triplet of electrons is prepared in the GHZM state which is
$\ket\psi = \frac{ \ket\uparrow \otimes \ket\uparrow \otimes \ket\uparrow - \ket\downarrow \otimes \ket\downarrow \otimes \ket\downarrow }{\sqrt{2}}$
The state is normalized. The relative sign of the two terms is negative. Each term is a tensor product of three factors. Reading from the left side, the factors describe the $$z$$-component of the spin of the electron on Earth, Jupiter, and Saturn, respectively.

Each of the three laboratories is able to measure the value of its electron's spin relatively to the $$x$$ axis or the $$y$$ axis. Each lab has to make 8 million measurements with those 8 million electrons that reach the lab. Each measurement is randomly (and independently from other labs) chosen to be a measurement of $$J_x$$ or $$J_y$$. You may see that the three labs will measure each of these 8 possible "arrangements" of the measured axes about 1 million times (plus minus a thousand or so):
$$xyy,\,yxy,\,yyx,\,xxx;\,\, yxx,\,xyx,\,xxy,\,yyy$$
The results of each of these 24 million individual spin measurements will be represented by the eigenvalue of $$\sigma_{\rm axis}=\pm 1$$ which is defined as
$J_{\rm axis} = \frac{\hbar}{2} \sigma_{\rm axis}, \quad {\rm axis}\in \{x,y\}$ We will be interested in "combined" measurements of the triplet of the electrons. I have already explained that there are 8 possible types of "combined measurements". They're given by observables which are simply products of the three $$\pm 1$$-valued outcomes for a spin:
\begin{align}
\end{align}
The superscripts 1,2,3 refer to Earth, Jupiter, Saturn while the subscripts indicate the axis. I added the crossed squares ($$x$$) and the empty or dotted ones ($$y$$) for a more readable version of the $$x/y$$ subscripts in the products. I had to eliminate the squares and filled squares because Internet Explorer's MathJax switches to black image fonts.

Similar operators $$E,F,G,H$$ with $$x$$ and $$y$$ interchanged are analogous but less interesting. The reason is that the values of $$A,B,C,D$$ are exactly predictable, as I will explain momentarily. On the other hand, the measurements of $$E,F,G,H$$ yield the results of $$+1$$ and $$-1$$ with 50% probabilities, so they're completely uncertain.

Those 4 million combined measurements in which we measure $$A,B,C$$ or $$D$$ are very interesting because we obtain $$+1,+1,+1,-1$$, respectively. That's because our GHZM tri-electron state $$\ket \psi$$ is an eigenstate of all these four operators. More explicitly, we have
\begin{align}
A\ket\psi &= {\bf \sigma^1_x} \sigma^2_y \sigma^3_y \ket\psi = + \ket\psi \\
B\ket\psi &= \sigma^1_y {\bf \sigma^2_x} \sigma^3_y \ket\psi = + \ket\psi \\
C\ket\psi &= \sigma^1_y \sigma^2_y {\bf \sigma^3_x} \ket\psi = + \ket\psi \\
D\ket\psi &= {\bf \sigma^1_x \sigma^2_x \sigma^3_x} \ket\psi = - \ket\psi
\end{align}
The state $$\ket\psi$$ is an eigenstate of each of these four "product" operators – which means that they have a totally predictable outcome – even though $$\ket\psi$$ isn't an eigenstate of the individual factors which means that each of the individual single-spin measurement has 50%:50% odds of producing $$\pm 1$$ as the result.

The fact that $$\ket\psi$$ is an eigenstate of each of the operators $$A,B,C,D$$ with the indicated eigenvalues isn't hard to check. It boils down to the fact that $$\sigma_x$$ and $$\sigma_y$$ act on the $$\ket\uparrow$$ and $$\ket\downarrow$$ states as follows:
\begin{align} \sigma_x \ket\uparrow = \ket\downarrow,\qquad \sigma_y \ket \uparrow &= +i \ket\downarrow\\ \sigma_x \ket\downarrow = \ket\uparrow,\qquad \sigma_y \ket \downarrow &= -i \ket\uparrow \end{align}
So both $$\sigma_x$$ and $$\sigma_y$$ exchange the two basis vectors and may add extra powers of $$i$$, too. Because each of the three spins is being acted upon by $$\sigma_x$$ or $$\sigma_y$$ in each of the operators $$A,B,C,D$$, it means that each of the operators $$A,B,C,D$$ effectively interchange the two terms in the GHZM state, $$\ket{\uparrow\uparrow\uparrow}$$ and $$\ket{\downarrow\downarrow\downarrow}$$. The fact that $$\sigma_y$$ with the extra powers of $$i$$ is included an even number of times in each of $$A,B,C,D$$ means that we do preserve the reality of the coefficients of $$\ket{\uparrow\uparrow\uparrow}$$ and $$\ket{\downarrow\downarrow\downarrow}$$.

One may also easily see that $$D$$ which only contains $$\sigma_x$$ simply interchanges $$\ket{\uparrow\uparrow\uparrow}$$ and $$\ket{\downarrow\downarrow\downarrow}$$ but because they had a relative minus sign in $$\ket\psi$$, the eigenvalue of $$D$$ is $$-1$$ because we fail to produce this extra minus sign when we permute the two terms. On the other hand, the presence of two copies of $$\sigma_y$$ in the operators $$A,B,C$$ produces an extra factor $$(+i)^2$$ or $$(-i)^2$$ which is $$-1$$ in both cases and the eigenvalues of $$A,B,C$$ are therefore $$+1$$.

It means that in one million of measurements of the triplets of spins when the arrangement is $$A$$, we always get the product $$A=+1$$. The same holds for one million of measurements of $$B$$ which always produces $$B=+1$$ and one million of measurements of $$C$$ giving $$C=+1$$. In one million of measurements of $$D$$, we obtain $$D=-1$$. Let me also remind you that $$D=-1$$ means that an odd number (1 or 3) of the three spins give the result $$-1$$ while $$C=+1$$ means that an even number (0 or 2) of the individual spins gives $$-1$$.

The combined properties are determined; the individual spins are not

It's important to emphasize that the state $$\ket\psi$$ is always ready to give predictable and universal outcomes when we measure $$A,B,C,D$$ – which is one million of combined experiments in each of the four cases. On the other hand, even though $$A,B,C,D$$ are constructed as products of three factors, three signs, each of these individual signs (individual spins) is undetermined, having 50%:50% odds for either result. All these claimed predicted by quantum mechanics have been experimentally confirmed.

Before each individual spin measurement, we know that the result will be $$+1$$ or $$-1$$. We also know that the group of three electrons must be prepared to produce the combined results $$+1,+1,+1,-1$$ whenever the three distant experimenters decide to measure $$A,B,C,D$$, respectively.

According to any realist theory that also respects locality, i.e. one that prohibits instant communication between Earth, Jupiter, and Saturn, the values of $$\sigma_{\rm axis}^{1,2,3}$$ to be measured must "objectively exist" right before the measurement. Imagine that each of the three spins is prepared to produce a particular result, $$\pm 1$$, for $$\sigma_x$$ as well as $$\sigma_y$$ measurements. If that were so, you could treat $$\sigma_{x,y}$$ as "classical observables". If they were classical observables, you might easily prove the following identity:
$ABC = +D$ Why is it so? Look how $$ABC$$ acts on each of the three electrons. It acts with a single copy of $$\sigma_x$$ and two copies of $$\sigma_y$$. The latter cancel because $$(\sigma_y)^2=+1$$ – after all, it's the same sign applied twice – and what you end up with is the single $$\sigma_x$$ for each of the three electrons. Their product is what we called $$D$$.

However, quantum mechanically, $$\sigma_{x}$$ and $$\sigma_y$$ are not classical numbers: they're observables and, for the same electron, they actually anticommute with each other! Note that each two operators from the list $$A,B,C,D$$ commute with each other because the number of permutations of $$\sigma_x^i$$ and $$\sigma_y^i$$ is even (two) for each of the 6 pairs. Because of this anticommutation rule for $$\sigma_x$$ and $$\sigma_y$$, we also have
$ABC = -D$ The extra minus sign appeared because we needed to permute $$\sigma_x$$ with $$\sigma_y$$ acting on the same electron an odd number of times before we were able to "cancel" the pairs of $$\sigma_y$$. If you want some details, then note that pairs of $$\sigma_y^1$$ and $$\sigma_y^3$$ could have been accumulated and replaced by $$1$$ without such sign flips. But $$\sigma_y^2 \sigma_x^2 \sigma_y^2$$ included in the product $$ABC$$ is equal to $$-\sigma^2_x$$ because one transposition of anticommuting Pauli matrices is needed.

This extra minus sign on the right hand side is what allows $$A,B,C$$ to have positive eigenvalues and $$D$$ to have a negative eigenvalue when acting on the common eigenstate $$\ket\psi$$, the GHZM state. But let me return to the classical logic. If it were true that $$ABC=+D$$, and the "real existence" of the values of $$\sigma_{x,y}^{1,2,3}$$ is a sufficient condition needed to "prove" that $$ABC=+D$$, then it would predict that if the triplet of electrons is ready to produce $$+1$$ for each of the combined measurements $$A,B,C$$, it must inevitably produce $$+1$$ for the combined measurement $$D$$, too. But it doesn't: experiments confirm the quantum mechanical prediction that $$D=-1$$ when acting on $$\ket\psi$$.

Local realism gives a wrong prediction for $$D$$ in 100% of the cases. Quantum mechanics with its new foundations is needed. You don't like it? Here is a recommendation for you:

What is needed to emulate quantum mechanics

Many people are unable to see or unable to admit that experiments that verify quantum mechanics (for example the experiment above) unequivocally rule out the idea that all the predictions of the experimental results may be explained by an independent "reality" that is certain and that exists prior to the measurement.

If we assume that $$\sigma_{x,y}^{1,2,3}$$ have "prepared" values to be measured right before each individual measurement, we may prove that the "prepared" value of $$D$$ is inevitably equal to the "prepared" value of $$ABC$$, so if the product of the guaranteed values of $$A$$, $$B$$, and $$C$$ is equal to $$+1$$, and it is the case for the GHZM state, $$D$$ must be guaranteed to yield $$+1$$, too. But it doesn't. Not only it "sometimes" fails to produce $$+1$$. The operator $$D$$ actually produces exactly the opposite value, $$-1$$, whenever we measure $$D$$.

It means that if you want to preserve the idea of some "realism", you must abandon the assumption of locality (which won't be enough, as argued below: I have to say it in advance). The individual spins are actually not encoded in some "local variables". There has to be a communication which guarantees that the value of an individual spin $$\sigma_{x,y}^{1,2,3}$$ that you measure depends on what other experimenters in the Solar System just decided to measure and what outcomes they received.

If this were true, you would of course have to give up relativity because there would now be fully physical superluminal signals instantly propagating through the Solar System. I designed the position of the three laboratories so that they were spacelike separated: not even light was fast enough to exchange the information.

But of course, the people who are eager to deny quantum mechanics, whatever the costs are, are also eager to deny Einstein's relativity when it's needed. And indeed, it's needed. Fine. So they may believe that quantum mechanics is fundamentally wrong – it just appears to be correct – and in the same way, relativity has to be an illusion, too. At the fundamental level, objects do communicate by signals that are not just "slightly faster than light" but they are arbitrarily fast. All the major sharp qualitative insights that have been found in physics during the 20th century may be just illusions – and there is some mundane classical "mechanism" that is faking quantum mechanics and that is faking relativity.

What would it really mean if the measurement of a single electron's spin depended on everything in the Solar System that has occurred so far, including measurements done on another planet a picosecond ago? If you really believe this thing – and it is on par to the belief that Young Earth creationism is fundamentally correct and the fossils are just a conspiracy to make the world look different than what it actually is – then you inevitably have to assume that there is a preferred reference frame in which the "real material physics" is actually taking place.

Whenever the three folks measure the three spins of the triplet of electrons, there must be an ordering which says that the measurement on the first two planets (and their results) determined the outcome on the third planet whenever the three measurements turned out to be of the kind $$A,B,C,D$$. According to relativity, the ordering of the 3 measurements depends on the reference frame. From the viewpoints of 6 distinct comets that fly in different directions, all 3!=6 orderings of the three measurements are equally good.

However, you – a diehard realist – must choose one objectively preferred ordering of the three measurements.

And the final, third measurement must get the information – instantaneously – and this infinitely fast counterpart of DHL is needed for... Why is it needed? It's needed for you to fake the predictions of quantum mechanics. Quite generally, I am amazed that you don't realize – or you don't care – that all your "science" is just trying to fake the truth by the untruth. This attitude of yours, namely attempts to find arbitrarily contrived "mechanisms" whose only goal is to deny something we almost directly observe – could be used in any other context in science and outside science, too. You may deny that people have visited the Moon, that the species have evolved, that the Earth has been here for billions of years, that the events in the Middle Ages actually took place. With enough dishonesty, you may deny anything.

Is there an emulator that mimics QM perfectly?

Let me assume that the reader understands that the success of the quantum mechanical prediction for a particular entangled three-spin state, the GHZM state, isn't a coincidence. Quantum mechanics has been tested in such a wide variety of situations that it's self-evident that its statistical predictions will work for any arrangement of elementary particles and spins. I think that the people who are trying to deny the postulates of quantum mechanics – as identified by the Copenhagen school – kind of know that any prediction of their "alternative theories" that differs from the quantum mechanical predictions by a finite amount will be falsified, much like lots of previous attempts to show that quantum mechanics was incorrect.

Great. So what you really want to do is to fake quantum mechanics perfectly. Can you do it?

You may hope that the answer is Yes but from the beginning, it's clear that you need to make billions of particular choices that don't really exist in Nature and that have no observable consequences. And for you to isolate the "faked quantum mechanics" inside a more general class of similar theories, most of which don't fake any quantum mechanical theory, you will need to adjust an infinite number of parameters. You will need an infinite amount of fine-tuning. Just to be sure, the amount of dishonesty you need to claim that such a proposed theory is a "real alternative" to proper quantum mechanics is much larger than the amount of dishonesty that a Young Earth creationist needs to dismiss the importance of fossils. The creationists only needs to "overlook" or "reinterpret" a finite number of fossils. You need to claim that infinitely many quantities predicted to be almost certainly nonzero by your non-quantum, non-relativistic theory (all the observable consequences of preferred inertial systems and material collapses) just happen to be accidentally zero. This denial of everything that physics knows about the natural phenomena is needed for you to protect your pet medieval belief in realism.

But the anti-quantum bigots have an unlimited reservoir of dishonesty so they're ready to go this path, too. What will they do? Obviously, they need to say that there is a real "wave function" or "density matrix" that should be treated in the same wave as a classical electromagnetic wave. Then there are some GRW-like collapses that make the world encoded in the "wave function as hidden variables" look like the real one. The wave functions must sometimes "literally collapse" and they suddenly shrink "in a particular reference frame".

You need to make the collapse physical and because the wave function quickly "shrinks" in this mode of reasoning, you need to specify a particular reference frame in which it "shrinks". Aside from choosing a particular reference frame, you will also have to choose a particular border above which things "really behave classically". I say it's artificial as well because according to quantum mechanics, the same quantum mechanical rules fundamentally apply to all systems, and not just the small ones.

When you analyze the GHZM experiment above, you will notice that the de Broglie-Bohm pilot wave theories are completely useless. They're only good for faking non-relativistic quantum mechanics with a fixed number of continuous degrees of freedom, like the non-relativistic quantum mechanical description of a single spinless particle. They're useless for spins. As we have discussed, you will inevitably get wrong predictions if you assume that the three spins have well-defined prior to the coordinate triple measurement.

The Bohmian theories say that the wave function "objectively exists" but it is just a "pilot wave" that drives another "objectively real" degree of freedom, the particle's actual coordinate and velocity. However, in the case of the spin, the GHZM experiment unambiguously shows that the "actual value of the spins" can't exist in the classical sense, so the particle-like portion of the Bohmian degrees of freedom or "beables" have to be thrown away, anyway. If you want to describe spins of particles, you may only use the "wave function" part of the Bohmian "beables" and this wave function spreads just like any other one.

Quantum mechanics has no problem with the spreading of this wave function because the wave function only encodes a half-baked complexified probability distribution which should be interpreted as the state of someone's knowledge, not as objective reality. If you want to say that the wave function describes some features of the objective reality, you have to offer some actual material mechanisms that "keep it from spreading". In the articles about the GRW collapse theories, including the recent one about Steven Weinberg, I've explained that any mechanism of "collapses" that you add into your theory to produce big enough effects to make the "objective wave function" more classical will inevitably introduce new perturbations that are safely excluded experimentally. So you will fail, anyway.

At the end, I think that even with the unlimited but finite amount of dishonesty, you simply can't construct a "realist" model that will fake quantum mechanics. The ultimate problem is that observers are quantum mechanical systems as well – and they may interfere with themselves. When a superior, high-precision observer A watches inferior observer B who watches system S, then B may think that she or he collapsed the wave function for S and everything becomes "objective" at some moment. After all, B arrogantly believes that her or his perceptions must be objective, doesn't she?

However, A is more accurate so he or she continues to describe B and S by a wave function that has several "macroscopically different components". A must do it because he or she knows that different portions of the wave function for B+S may still be able to interfere in the future. The "precision standards" adopted by A mean that the collapse occurs much later according to A than when it occurs according to B. All conceivable discrepancies between the interpretations of the history by A and B are faults of B and they may be explained as imperfections of her brain or logic.

This is a part of a more general theme: the "moment" when the collapse occurs is completely subjective.

This statement has many aspects. The simplest one to understand was mentioned at the beginning: if there were a "real collapse", it would mean that the wave function of a particle that was just absorbed must "instantly disappear" from the rest of the space. However, the term "instantly" requires you to specify a particular reference frame because special relativity guarantees that "now" has a different meaning for observers in different states of motion.

When you look how Nature actually works, i.e. when you study these issues quantum mechanically, the reference frame in which the wave function collapses "now" is completely unphysical because the portions of the wave functions that "disappear" can't influence anything that latter happens, not even in principle! They just describe some a priori possible outcomes that didn't materialize; they only exist in the head of a physicist who calculates the predicted probabilities. When a different outcome becomes a fact, we have to switch to conditional probabilities so the outcomes violating the corresponding conditions just don't affect any future predictions or outcomes. By the very definition of conditional probabilities, the influence of the later-disappeared portions of the wave function on any observable phenomena is exactly zero, something that wouldn't be possible in any natural "realist" theory.

A few paragraphs ago, I also mentioned another aspect of this "subjectivity" of the collapse: less accurate observers may want to "perceive an outcome of a measurement" and they may do so prematurely. More accurate observers are calculating things more accurately so they still describe the rest of the world, including the sloppy/smaller observers from the previous sentence, by superpositions, so that they don't forget about their potential to interfere. In quantum mechanics, this aspect of the subjectivity of the collapse is unsurprising and free of problems, too. The "collapse" just means a subjective process by which "a physicist is taking recently observed facts into account" (he modifies his state of knowledge about the world) and it doesn't matter when he does so. Different observers predicting the same measurable final outcomes may do so at different stages and they will still agree about the predictions of events that they may consider as shared facts. These collapses don't have to have objectively unique properties, moments, and shapes – and indeed, they demonstrably don't: they're a part of a subjective cannon allowing physicists to make valid probabilistic predictions.

Any "realist" description of the collapse must inevitably choose a point beyond which the collapse is "strictly real" and the potential for interference is lost after that point. But in the world around us, as described by quantum mechanics, this never happens. In principle, it is always possible to think about a more accurate observer who calculates the motion of all particles – including us – using the exact wave function with arbitrary superpositions etc. Any "realist" picture of the "wave function collapse" must introduce a new "bureaucratic intervention" after which the world is "obliged to behave classically". For the GRW collapse proponents, this intervention is the "flash" that shrinks a wave function. For the many-worlds advocates, it's the moment when you "objectively split the world" into many.

But according to quantum mechanics, such events when the properties of the physical systems become "exactly classical" never occur. As we study larger objects, the classical approximation becomes more tolerable, but it is never fundamentally right. So the collapses and the events in which the many worlds split never occur. And they don't need to occur because the amplitudes only have a probabilistic interpretation and probabilistic distributions are allowed to spread without the underlying truth's becoming fuzzy.

What I finally want to emphasize is that all this redundant and "objectively real but totally unobservable" superstructure – from many worlds to extra invisible Bohmian positions of particles (which can't help in the case of spin or particle production, anyway) or other hidden variables to GRW collapses prescribed from above – is only being invented because certain people behave as bigots who are unable to admit that the physics research in the 20th century has irreversibly falsified all intrinsically classical models of the reality. All the new "fanciful stuff" with tons of choices and processes (superluminal communication, preferred frames, collapses, the length scale to which the GRW collapses shrink the wave function, the frequency of such flashes etc.) that can never be observed and with the infinite amount of fine-tuning and obfuscation that is needed for it to fake the real, relativistic quantum world (to guarantee that none of the new predictions is really observed) is only being proposed because some people's bigotry has no limits. Their dogmas about "realism" are more important for them than any amount of empirical evidence, more important for them than everything that science has actually found.

People, those bigots who are still denying that the insights of quantum mechanics inevitably force one to be careful about the positivist, instrumentalist principles and that forces us to understand science as a gadget to organize our observations rather than to promote the idea about an "independent real world": give up, apologize, shut up, and calculate!

And that's the command.

1. This question may be philosophical, but it's serious, so I don't think I deserve to be banned. Lubos is always annoyed with those who don't accept quantum mechanics, and rightly so. But what about those who accept it and find it disturbingly strange? For instance, if there is no 'definite external reality' until it is observed or measured, then is the universe a figment of my (or your) imagination? This is a fair question asked by many good physicists, not just Einstein.

2. It's not so strange, Benjamin, if QM are viewed as a phenomenological description of the world with a treatment of "errors of observation" built right into it.

Continuum mechanical descriptions of the world are verified only by measurements, these measurements always involve errors, when indeterminate errors propagate in a certain way we conclude that the continuum description is an accurate (or adequate) reflection of reality.

Same with QM, except one needs to interpret carefully how "errors of measurement" must propagate in a quantum interpretation.

To me anyway, the world would be a lot stranger if the quantum of action h was exactly zero. How could an atom be constructed if this was true? How would EM radiation of heated objects be distributed amongst various wavelengths?

I have no idea, I don't think atoms as we know them could exist

3. Did you miss out on the lecture from earlier this year where he asked us to consider the scenario where the police know the starting location of a criminal, and that the possible future location of that criminal spreads out like a wave over time due to the criminal's unknown speed and direction?

Once the police finally do pinpoint the criminal's location at some point the future, the wave instantaneously collapses.

It's common sense, and does not involve voodoo magic.

4. It is possible to produce correlations and anticorrelations (and so on, and so forth) between events in space-time if one adopts Huw Price's view that the state of a particle is constrained by what occurs at both ends of its worldline, rather than having the past end somehow influence the future end. This is because at the level of particles (or whatever those quantum entities really are) there is no entropy gradient, no "arrow of time". Or rather the arrow points in both directions with equal force. It's only the fact that we macroscopic beings, experience the entropy gradient so strongly that we frequently forget this. However, this means that one cannot point to the outcome of a quantum measurement and say "That couldn't happen according to classical views because of causality!" Because at that level, the laws of physics are time-symmetric (this is an observed fact with the exception of CPT violation in some weak interactions).

Bringing this extra fact to the table, as it were, means that a lot of "quantum weirdness" evaporates. For example, you can explain EPR correlations quite easily if you allow that the "final" (measured) states of the particles will have an influence on their "initial" state (for example, on how a pair of photons are emitted - or we could equally say, absorbed! - by a laser).

However, there are quite a few "diehards" who insist that the above just ain't so. They object to things "going back in time" and so on, missing the point that such talk is only applicable to macroscopic entities and irrelevant to the (as far as our current level of understanding of science goes) uncontentious fact that the laws of physics are, at the quantum level, time-symmetric.

I would be interested to know how the above influences the conclusions you've drawn about the GHZ experiment, if at all.