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Has Anton Zeilinger created a time machine?

Abstract of this blog entry: No.

I am convinced that Anton Zeilinger understands quantum mechanics, the new rules according to which the world works. However, he has become famous for being a magician of a sort – someone who benefits out of fooling people – and it seems to me that he enjoys it.

His being a linear superposition or entanglement of Niels Bohr, Criss Angel, and David Copperfield is nicely documented by his group's new paper in Nature Physics,

Experimental delayed-choice entanglement swapping (arXiv copy)
The last sentence of the abstract – and it's very far from being the only sentence in the paper with a similar message – says that "[t]his can also be viewed as 'quantum steering into the past'". Just turn the steering wheel around and travel to the past. Oh, really? ;-)



The reason why Zeilinger et al. partly act as magicians is that much of their paper is attempting to impress the reader by pointing out that something "looks like something" paradoxical. (The same comment applies to those atomic physicists and opticians who made "something in their lab look like it moves superluminally" and many others.) Except that there is nothing paradoxical about the experiment as one may confirm if he looks carefully and rationally.

I will discuss the experiment right after the break.




As the title indicates, the experiment combines two things. One of them is the virtue of the "delayed choice" – the fact that some decisions are made later than in "immediate choice" of these experiments to make the results look more confusing to the laymen (although people who understand quantum mechanics immediately see that quantum mechanics self-evidently guarantees that the results won't depend on the moment of the choice so we don't learn anything new whatsoever by making the experiment a "delayed choice" one). See Delayed choice quantum eraser to understand what the delayed choice is all about and why some people are so obsessed with it.

The second ingredient that the paper incorporates is entanglement swapping. I will start my technical explanations with this concept. Entanglement swapping is pretty much the same operation as other operations that have been given fancy names such as "quantum teleportation" etc., one only uses two pairs of qubits. Consider a four-photon state\[

\ket{\Psi}_{1234} = \ket{\Psi^-}_{12} \otimes \ket{\Psi^-}_{34}

\] where\[

\ket{\Psi^-}_{12} = \frac{\ket H_1 \ket V_2 - \ket V_1\ket H_2}{\sqrt 2}

\] is an antisymmetric, maximally entangled "singlet" state, and similarly for the photon pair \(34\). You see that the pair \(12\) isn't entangled with the pair \(34\): we have a tensor product, after all. However, one may measure some correlated properties of the photons \(23\) which, due to the entanglement between \(1\) and \(2\) as well as between \(3\) and \(4\), automatically entangles the pair of photons \(14\) as well.

This entanglement swapping is trivially predicted by quantum mechanics (it is totally redundant to do their actual experiment because its results were safely predictable already in the mid 1920s) and is used in various quantum cryptography protocols etc. The only thing that those folks add in the new paper is the "delayed choice" aspect: they look at the photons \(14\) after the choices on \(23\) have been done and after the \(23\) photon pair has actually disappeared. Much like in the case of "delayed choice quantum eraser", it may "look like" someone is rewriting the past. And because Zeilinger et al. are partly magicians, they love to nurture this misconception of the consumers of their work.

Their Figure 1 makes the point of the experiment very clear:



Start from the bottom. The photons \(12\) are produced in the EPR entangled pair and so are the photons \(34\) but the groups \(12\) and \(34\) are independent of each other. Alice and Bob measure properties of the photons \(1\) and \(4\). They decide to make one of the three kinds of measurements (both Alice and Bob use the same kind) and for each of these three choices, there are two possible outcomes for Alice and the same two possible outcomes for Bob:\[

\ket H / \ket V, \qquad \ket R / \ket L , \qquad \ket+ / \ket -

\] Ignore the middle photons \(23\) on the diagram. You see that on every line, Alice and Bob measure the same property (one of the three possibilities above) but their results are completely uncorrelated because the groups \(12\) and \(34\) were not entangled. The standard formulae of quantum mechanics clearly imply there's no correlation. Let me just state explicitly that the measurements that Alice and Bob perform are either measurements of the linear polarization (vertical or horizontal?), or the measurements of the circular polarization (left-handed or right-handed?), or the measurements of the linear polarizations along the 45° axes (\(\ket\pm\), i.e. Southeast or Northeast).

However, there's a Victor in the middle. He measures the photons \(23\). Note that the Hilbert space describing these two photons is 4-dimensional (\(2\times 2 = 4\)) so one may find a basis consisting of 4 vectors. The authors spend a lot of time by emphasizing the differences between the four basis vectors – which of them are entangled and which of them are not – but this is just a way to fool the readers (and maybe some of the authors themselves). To discuss these matters rationally, it's legitimate to simply list the orthogonal basis that Victor uses for the two photons in the middle:\[

\ket V\otimes \ket V,\quad \ket H\otimes \ket H,\quad \ket{\Phi^+}, \quad \ket{\Phi^-}

\] where\[

\ket{\Phi^\pm} = \frac{\ket H_2 \ket H_3 \pm \ket V_2\ket V_3}{\sqrt 2}

\] I suppose there's an error in the paper and \(\Phi_{23}\) should be \(\Psi_{23}\) because the photons \(23\) may also be found in the \(\ket{HV}\) or \(\ket{VH}\) states which have a vanishing probability for any outcome of the measurement that Zeilinger et al. announce. Alternatively, what they mean is that Victor chooses whether he measures the horizontal/vertical or \(\ket{\Psi^\pm}\) properties of the photons \(23\) and eliminates all pairs of photons \(23\) for which the outcome is something else than the options above i.e. all measurements in which he gets either \(\ket{VH}\) or \(\ket{HV}\) or their combinations such as \(\ket{\Psi^\pm}\).

The details modify the exact discussion of the experiment but they don't matter for the overall picture. What may be derived from quantum mechanics are the probabilities of various arrangements of results seen by Alice, Victor, and Bob. In particular, if Victor measures \(\ket{VV}\) or \(\ket{HH}\) on photons \(23\), it guarantees that these photons are "in the same state", and because of the internal entanglement in the pairs \(12\) and \(34\), it also implies that the photons \(1\) and \(4\) seen by Alice and Bob will reveal the same outcome. In other words, \(1\) and \(4\) will now be measured as entangled, inheriting the entanglement from their entangled partners \(2\) and \(3\) "who" may already be dead. That's the entanglement swapping but if you didn't know these fancy words and you just used the standard linear algebra of quantum mechanics, you would miss no physics whatsoever.

If Victor measures the photons \(23\) to be in an entangled state such as \(\Phi^\pm\), one may calculate that the results for the photons \(1\) and \(4\) have 50% odds to coincide and 50% odds to differ and surprise, that's what the measurements found, within the error margin, as well. No correlations between \(1\) and \(4\) in this case.

It is not terribly important to go through one particular exact definition of a similar experiment and/or to do this experiment in practice. It's totally clear that the experimental results will agree with the quantum mechanical predictions. And it's also totally clear that the quantum mechanical predictions – which are easily calculated because they only involve the decomposition of a known ket state to one of the bases of the 16-dimensional Hilbert space for four photons' polarizations – imply that it doesn't matter a single bit when we measure the polarization of either photons. The polarization of each photon is conserved after the photons are prepared – in the sense of vanishing derivatives in the Heisenberg equations of motion – so the probability distributions for individual outcomes manifestly don't change with time.

So why do the people feel the urge to talk about "steering to the past" and similar bullshit?

One isn't steering into the past at all. At most, one is inventing silly fairy-tales about Victor's ability to rewrite the past. By seeing some property of the photon pair \(23\), Victor may give a "new purpose of life" of the photons \(14\) seen by Alice and Bob and those photons \(14\) may already be absorbed for quite some time. But physics isn't about inventing fairy-tales about the purpose of life of two photons. From the physics viewpoint, the life of two photons has no purpose. The purpose of physics is to make predictions of the experiments and one can't improve his ability to make predictions by learning something after you actually need the predictions.

As always in these discussions, quantum mechanics implies that there are correlations between the measurements by Alice, Victor, and Bob, at least in some cases – when they measure some particular properties of individual photons or combined properties of photon pairs. But this correlation doesn't imply a particular causation. In fact, there is a particular causation in all these experiments that is very different from the fairy-tales that many people are trying to propagate: the actual cause of all the observed correlations is that the four photons were prepared in a particular state, \(\ket{\Psi_{1234}}\).

The reason behind all the patterns and correlations and enhanced or reduced probabilities of anything in quantum mechanics is in the initial state.

The idea that Victor "affected" the measurements by Alice and Bob – measurements that could have been done in the past – may only result from some attempt to create a classical model of what's happening in our quantum world. But our world isn't a classical model so all "impressions" one gets by looking at a classical model that disagree with the statements by quantum mechanics are just wrong. Quantum mechanics makes it clear that not only a particular measurement can't influence the past or the probabilities of different outcomes in the past; the measurement or any other process can't even influence events that are spacelike-separated.

In the last two sentences before the acknowledgements, the Nature Physics paper says:
This formation of subsets is independent of the temporal order of the measurements. According to Wheeler, Bohr said: “No elementary phenomenon is a phenomenon until it is a registered phenomenon.” [5, 7] We would like to extend this by saying: “Some registered phenomena do not have a meaning unless they are put in relationship with other registered phenomena.”
Nice. It's also important to mention that there is one big difference between the statement by Bohr and the "bonus" statement that Zeilinger et al. add. The difference is that Bohr's statement is a valid, key cornerstone of quantum mechanics, an insight about the actual natural phenomena around us, while the "addition" by Zeilinger et al. is pure rubbish.

Bohr is saying the usual thing that in quantum mechanics, one can't consider properties of objects to be classical facts before they're actually measured. For example, in Feynman's path-integral approach to quantum mechanics, one must be summing the complex amplitudes over all conceivable histories and avoid any claims that one of the histories actually took place. Such a claim would lead to wrong predictions. Things must be allowed to interfere and do whatever they want and what they want doesn't admit any classical description. Only when they're measured, they may turn into classical facts.

What Zeilinger et al. are adding is just a delusion. The measured spins or other properties in Alice's and Bob's notebooks may fail to have a "meaning" in the sense of a religious "purpose of life" or something like that. But physics isn't about those things. They still have a meaning as pieces of information. As soon as Alice and Bob measure the spins, they acquired some pieces of information that may be considered by them as classical facts.

Quantum mechanics predicts and experiments confirm that the spin measurements by Alice and Bob are uncorrelated and that's the only valid thing one may say if we only look at the measurements of Alice and Bob only. However, quantum mechanics also predicts that there is a correlation between the outcomes seen by Alice, Victor, and Bob. So if we look at the results seen by all these three folks, we will see that they're not independent. Some of the combinations of the results are forbidden – they have a vanishing probability that may be calculated from the initial state. Much more generally, quantum mechanics allows us to predict the probabilities (frequencies assuming that we repeat the experiment many times) of any possible alternative outcomes for any combination of "types of measurements" that the experimenters may choose.

By choosing his own measurement, Victor isn't giving any additional information to Alice and Bob that would allow them to improve the predictions for their own measurements. It's too late for such improved bets because Alice and Bob have already made the measurements. Instead, by making an experiment according to his own choice, Victor is only verifying the correlations predicted for Alice-Victor-Bob by quantum mechanics. If Alice and Bob make their measurements before Victor and if he learns about the outcomes, he should use the "conditional probability" that takes the information obtained by Alice and Bob into account. In the misleading laymen's vocabulary, Victor should "collapse the wave function" for his photons \(23\) before he makes a prediction for his photons. In this "materialist" interpretation of switching to conditional probabilities, Alice's and Bob's measurements affected Victor's measurements, not vice versa. Even with a "materialist" interpretation of the "collapsing wave function", one may always describe the situation in such a way that the past data constrain the current or future ones and no retrocausal influences ever occur.

The "magical trick" by which Zeilinger et al. want to fool you is their implicit claim that Alice, Victor, and Bob are not just three "equally important" subsystems of a larger system for which quantum mechanics predicts probabilities of outcomes including all the correlations. Instead, they want you to believe that Victor is the "Master of the Universe" whose decisions are more important than all other phenomena in the Universe, both in the future and the past, and this Victor is "controlling" everything else. But this ain't the case. Victor doesn't control anything.

I have discussed the very same things in my recent text on nonlocality (and very similar things in many other entries) but it may be a good idea to try to repeat these elementary things once again, in the context of Alice, Victor, and Bob.

The emotional reason why Victor is being presented as the "Master of the Universe" (without a glimpse of a rational justification) is that he's the only one among the three who is making a decision in his own mind. But one could also design more complex experiments in which the other folks would be making their decisions as well. Would they be also Masters of the Universe? Who wins if several Masters of the Universe fight?

Of course, in reality, none of them affects any property of a remote system and surely no property in the past. Even if you were imagining that Nature is a hot babe who "literally throws dice", to make sure that the outcomes are random but statistically agree with the QM-predicted distributions, She doesn't need to know anything about Victor's (future) decisions when she's deciding about the outcomes that should be seen "right now" by Alice and Bob. The measured outcomes in Alice's and Bob's labs are given purely by the probabilistic distributions for the photons \(1\) and \(4\), i.e. by some \(\rho^{(14)}\) of a sort, and this object is easily shown to be independent of the basis for photons \(23\) that Victor picks. Why?

All the probabilities that Nature needs to throw dice and produce random results for Alice and Bob may be calculated from the density matrix \(\rho^{(14)}\) for the photon pair \(14\). For example, the probability that we see a property associated with a projection operator \(P\) is simply \({\rm Tr}(\rho^{(14)} P)\); this is actually enough for everything Nature needs to know. And if we have a pure state for all photons \(1234\) like in this case, the density matrix for \(14\) may be obtained as a partial trace:\[

\eq{ \rho^{(14)} &= {\rm Tr}_{2,3} \rho^{(1234)} =\dots\\
\dots &= {\rm Tr}_{2,3} \ket{\Psi}_{1234}\bra {\Psi}_{1234}=\dots \\
\dots &= \sum_{i=1}^4 \braket{b^{(23)}_i}{\Psi_{1234}}\braket{\Psi_{1234}}{b^{(23)}_i}
}

\] where the final sum goes over an orthonormal basis of the Hilbert space for photons \(23\) and the two inner products in the final sum are just "partial" inner products that "annihilate" the indices corresponding to the \(23\) Hilbert space but preserve the indices for the \(14\) Hilbert space so that the resulting object is still a matrix acting on the \(14\) Hilbert space.

A basic fact about the linear algebra underlying quantum mechanics is that it doesn't matter a single bit which orthonormal basis \(\ket{b^{(23)}_i}\) for Victor's Hilbert space we use. This fact is known as the completeness relation:\[

{\bf 1} = \sum_{i=1}^4 \ket{b_i}\bra{b_i} = \sum_{m=1}^4 \ket{\beta_m}\bra{\beta_m}

\] In the previous displayed calculation, this relation was used in combination with the partial trace which is the reason why the order of the ket and bra vectors was switched.

That means that his decision "what kind of a measurement Victor chooses" makes no impact on Alice's and Bob's results (or the probability distributions for their photons and their correlations), either. It only affects what results the whole group Alice-Victor-Bob may obtain. But in this group (that collectively measures some correlations), Victor isn't a boss. He's just one member and any correlation between all the members that finally shows up in the data has been calculable and provable since the very first moment when someone prepared the state for all the four photons, i.e. from the very beginning.

Once again, Victor's personal decisions make no impact on the composition of the outcomes measured by Alice and Bob. They only have an impact – and one that isn't surprising at all – at the composition of outcomes that are measured by groups that do include Victor, for example on the composition of results obtained by the whole Alice-Victor-Bob group. And it shouldn't be shocked that the correlations in such groups may only be verified after all the members of the group complete their measurements.

Just like Victor's "choice of the basis" for the photons \(23\) makes no impact on the results measured with the photons \(14\), the timing of the measurements is inconsequential, too. The experiment is designed so that the polarizations of all the photons are totally preserved (as operators) from the very first moment when the initial state of four photons is prepared. It follows that the probabilistic distributions for individual photons and their groups won't depend on the timing of the measurements, either. All these facts are trivially seen from the very description of the experiments and may be translated to completely rigorous and provable propositions in the quantum mechanical formalism.

One of the irrational things that I see behind these neverending games with basic properties of qubits is the continuing disbelief that entanglement exists. But entanglement is the "default condition" of degrees of freedom in the Universe. Whenever some degrees of freedom share a common past, i.e. whenever they were influenced by some common events in the past, they will be entangled to some extent (up to the measure-zero subset – which is nevertheless important – when the entanglement is zero). Entanglement is just a quantum description of the concept of a correlation. For most types of correlations in a quantum world such as ours, these correlations don't have any right classical model. So models involving classical correlations, with an objective reality that exists prior to the measurement, is inapplicable to almost anything in the real world.

This paper and other papers also make a big deal about "qualitatively different interpretations" one may give to various outcomes seen by Alice and Bob. But quantum mechanics implies that no such qualitatively different interpretations exist; it predicts the probabilities of any measured outcomes as continuous (and therefore only quantitatively changing) function of the initial state as well as the operators whose values are measured. Two of the special values of probabilities that may be predicted are 0% and 100% but they don't "qualitatively" differ from the nearby numbers. So all the talk about "qualitatively different interpretations" only means that some of them are easier to be reconciled with a "classical model". But classical models don't play any role in Nature which simply isn't classical, so also this separation of the outcomes to "qualitatively different groups" is bogus.

Papers including the current Zeilinger et al. paper in Nature Physics are clearly unwilling to appreciate this insight – that quantum mechanics is right and nonzero entanglement is the normal, default state of affairs – that's been acquired 85 years ago. So we're still bombarded by papers saying "look, it's amazing, there's another experiment that disagrees with classical physics and its intuitive understanding of correlations, evolution, and causality". However, almost everything that happens in the microscopic world (and not only microscopic world!) is incompatible with these components of the classical intuition. It's just very stupid to be "shocked" by this fact 1,000 times in a row because it shows that someone isn't able to draw lessons, someone isn't able to learn. However, I have already "partly" reconciled myself with the fact that people will never really swallow quantum mechanics so these irrational discussions will continue indefinitely. That's what I predict which doesn't mean that I am already happy about it!

And that's the memo.

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