## Thursday, March 08, 2018 ... /////

### No superstitions in the delayed choice quantum eraser

People should listen to arguments of different types than they expect or they're used to

I've noticed that lots of the people's annoyingly irrational, stubborn approach to rational arguments – that are relevant in very many diverse topics, from multiculturalism to the quantum entanglement – may be blamed on this tendency of theirs:

Just attack every kind of an argument that you're not used to, that you don't expect, and don't listen to it at all. Only the statements or arguments you are used to repeat should be repeated.
In effect, I am often talking to a wall. When I try to explain why the energy carried by a strictly periodic configuration is quantized in quantum mechanics, the recipient of the explanation just doesn't like the conclusion. So he often attacks every piece of your explanation by irrational fog and hostile chants, effectively pretending that you haven't made any argument (and sometimes, it's basically a complete proof) at all. He effectively assumes that he already knows everything even though he knows and understands almost nothing.

Even if one knows lots of facts, arguments, derivations, and proofs, there always exist additional facts, arguments, derivations, and proofs. To say the least, there exist other ways to look at a problem, other ways to deduce something new about the problem, other conclusions that may be deduced. And curious, impartial people are interested in those. But the people who just don't listen and assume that what they "know" is the only correct thing or the only thing worth knowing are just arrogant idiots.

OK, let me revisit the delayed choice quantum eraser because someone posted several comments about the article on that experiment that was written in 2016 (see also a 2010 blog post).

All these comments were conveying the opinion that the "future affects the past" [John caught the permutation, thanks]. Well, it doesn't. There isn't any retrocausal influence like that in that experiment or in (our quantum mechanical) Nature in general. The blog post's main point (and the title) is all about the absence of such retrocausality. But when someone is just obsessed with a delusion, he overlooks absolutely everything. He overlooks the proofs that he's wrong even if he's swimming in the sea of such proofs.

Again, what's going on in that experiment?

The sketch of the experiment is very complicated but the main reason why people have so much trouble with this experiment – or almost any experiment that deals with the information in the quantum way – isn't about some complicated technical details in the setup, I think. It's about their completely wrong thinking about certain things that are actually very straightforward.

This experiment combines a double slit experiment with an entanglement experiment in a certain way.

A photon goes through a double slit. Instead of being captured on a photographic plate immediately after that, the photon is transformed into a photon pair. The upper photon continues to draw an interference pattern and its position $x \in\RR$ within such an interference pattern is measured by the detector D0 at the end, while the lower photon (its partner) goes through beam splitters and mirrors and ends up being detected discretely in one of the detectors D1,D2,D3,D4.

To describe this evolution and detection quantum mechanically, you need some Hilbert space that contains states with one or two photons at different places, you need to know the evolution operator on that Hilbert space that includes the action of the double slits, Glan-Thompson prism (which splits a photon to a photon pair), the beam splitters, mirrors, and the detectors.

At the end, the position $x$ of the upper photon in the detector D0 is being measured; and it's being measured whether the lower photon landed in D1, D2, D3, or D4 – so a discrete number from the set $y\in\{1,2,3,4\}$ is measured for the lower photon. As soon as the original, single photon goes through the double slit and after that photon is split to the photon pair, quantum mechanics already makes a prediction for these possible measurements of $x$ and $y$ that could be done later.

Off-topic but hilarious: a 9-minute-long, 1-million-viewers-collecting film about the penetration of alternative, politically correct mathematics to a U.S. school. Don't worry, there will be a limited happy end. Hat tip: V. Dlab

OK, the predictions are probabilities or "a density of probability" which are calculated from Born's rule – the probability is the squared absolute value of a complex probability amplitude – from the complex probability amplitudes that are matrix elements of some linear unitary evolution operator. So let's assume that you know how to deal with the linear operators, complex probability amplitudes, and so on. When you analyze this experiment – which is a combination of the double slit and entanglement experiments – you end up with the distribution$\rho(x,y), \quad x\in\RR,\,\, y\in\{1,2,3,4\}$ that contains everything that may actually be measured in this experiment. It's a probability distribution – I used the Greek letter $\rho$ (rho) for a distribution instead of $p$ for a probability because $x$ is continuous. OK, what's the predicted distribution? I need to inform you about the function $\rho(x,y)$ but because $y$ has four (finite number) of possible values, the information in $\rho(x,y)$ is equivalent to four graphs of $\rho(x)$ obtained by setting $y$ to one of the four values. So these four functions of $x$ look like this:

I won't derive it in detail because it's straightforward. But you should see the final prediction and realize that it's utterly reasonable. The first graph among the four, one for $y=1$, looks like an interference pattern drawn by the upper photon. If you catch the lower photon in a particular state – one from which the location in the double slit (the "which slit" information) cannot be determined – the splitting by the prism is mostly immaterial, and you get a standard interference pattern.

The second graph among the four, one for $y=2$, is another interference pattern, one whose phase is shifted by $\pi/2$. So it has maxima pretty much where the first graph had the minima and vice versa.

Finally, the graphs for $y=3$ and $y=4$ – corresponding to the detection of the lower photon at D3 or D4 detectors – don't exhibit any interference pattern. It's because by the measurement of $y=3,4$, one effectively measured the "which slit" information of the parent photon (and therefore both offspring, as they emerged from the splitting prism). When you measure the "which slit" information, there can't be an interference pattern, and indeed, there is none.

Some of the most hard-working people among those who end up screaming "you may rewrite the past" understand everything that I wrote above. You may draw the graph of $\rho(x,y)$ by drawing four functions of $x$. Well, if you do so, you must first decide what is the discrete value of $y\in\{1,2,3,4\}$, and then you may draw the graph as a function of $x$. Because the measurement of $y$ – the detection of the lower photon in one of the detectors D1,D2,D3,D4 – may be done after the $x$ of the upper photon is measured, they decide that it means, in this particular chronology, that the later event, the detection at D1,D2,D3,D4, has influenced the earlier event, the drawing of the interference pattern by picking a value of $x$. So the future affects the past.

Except that this conclusion could have only been made because the person has assumed it. There is absolutely nothing that forces you to make this conclusion. Why? Because you don't have to slice the function $\rho(x,y)$ to four graphs by the horizontal cuts. It is equally possible to slice the function $\rho(x,y)$ into vertical cuts – which is appropriate for the chronology in which $x$ is measured before $y$. So just look at the graph of $\rho(x,y)$:

If you measure some value of $x$ first, just draw a "united" vertical line through this bunch of four graphs. When you do so, the vertical line will produce four intersections with the four graphs above. And the heights of these four intersections may be interpreted as $P_1,P_2,P_3,P_4$. These are nothing else than the probabilities that the lower photon will be measured as $y=1$ or $y=2$ or $y=3$ or $y=4$ – whether it will be detected at the detector D1 or D2 or D3 or D4.

So there is absolutely no reason to think that $y$ should be decided "first" and it's the "cause" of the measured value of $y$. Instead, the variables $x$ and $y$ play a pretty much totally symmetric role. The probability distribution that is a function of two (or more) variables may be sliced in any way – any order – you wish, despite the fact that some of the variables belong to a continuous spectrum and others belong to a discrete spectrum.

The retrocausal people assume that the decision of $y$ must be made first, or it's the cause. But they fail to realize that there's nothing that can justify this assumption, this particular asymmetry between $x$ and $y$. This assumption is completely arbitrary and rationally unjustifiable: variables $x,y$ that a distribution $\rho(x,y)$ depends upon are obviously "equally good" and none of them is the "only allowed master". These retrocausal people don't want to listen to this point. They don't want to think rationally. They have decided that they only want to consider the discrete variable $y$ to be the original choice and therefore the cause of other things, and the profiles that depend on the continuous $x$ must be the effect of the choice of $y$. There's no reason why it should be so and if you think rationally, you will see that I am right. But those people just don't want to think rationally.

In quantum mechanics, one deals with the wave function or the density matrix and at every moment, the wave function or the density matrix is "ready" for any kind of a measurement. By Born's rule, probabilities of one outcome or another of an experiment may be calculated. When a measurement is actually performed, the wave function or the density matrix "collapses" i.e. it's projected to the subspace of the Hilbert space that corresponds to the recently measured value of the observable that was observed. The first observable that is measured may be $y$ of the lower photon but it may be $x$ of the upper photon, too.

The setup of this experiment is designed so that the value of $y$ is basically conserved in time. You can make the paths in the lower part of the experiment as long as you want. The value of $y$ – i.e. the information about the detector (D1 or D2 or D3 or D4) where the lower photon will be detected – is conserved in time (the Heisenberg equation of motion is $dy/dt=0$) which means that it's completely irrelevant whether $y$ is measured before or after $x$ is measured! You may always trace your predictions for the coming experiments by following the measurements chronologically and collapsing the wave function according to the measurement in their actual chronological order. After some moment, when the two offspring photons get distant from each other, they don't interact with each other at all which is why it becomes totally irrelevant which of the two photons is measured first! The obsession with "which photon is measured first" is completely irrational. After all, according to special relativity, the answer can't even be independent of the chosen inertial system.

When quantum mechanics – or even a classical theory – predicts some distribution $\rho(x,y)$, it basically remembers some two-dimensional pattern that will be drawn by the photon pairs. And in this two-dimensional pattern, there is absolutely no reason to think that one of the coordinates has to be measured before the other one. They play an equivalent role. None of them is more "the cause" than the other, none of them is more "the effect" than its partner.

Not only there is no retrocausality. In realistic quantum mechanical theories (at least outside quantum gravity), there is no non-locality, either. As I have discussed in previous blog posts, the decisions "what to measure" made in the region A don't influence the probabilities that a particular outcome of any observable will be measured in distant, spacelike-separated region B. That's what guarantees that there was no influence made by the people in A on the objects in B.

Only when one knows an outcome of the measurement, e.g. an outcome in A, he may use the knowledge, substitute a value of $a$ into some probability distribution $\rho(a,b)$, and "collapse" this two-dimensional probability distribution $\rho(a,b)$ to a one-dimensional $\rho(b)$ where we simply substituted the recently measured value of $a$ into $\rho(a,b)$. This remaining distribution $\rho(b)$ will predict different probabilities in general for $b$ than $\rho(a,b)$ did before $a$ was known. That's because $a,b$ are correlated in general. But the correlation of $a,b$ isn't a consequence of the very recent measurement of $a$. The correlation between $a,b$ is a consequence of the events in the past that prepared $a,b$ as a composite system composed of two parts that have interacted (or that were born together). If they hadn't ever interacted or co-existed, they couldn't be correlated or entangled!

Also, the irrational person was insisting on getting a story about the photon that decides about some things while it's flying through the beam splitters. Well, the photon doesn't have a brain or other anthropomorphic features. And nothing is measured at the beam splitters – which are just dull parts of the setup. So there are no probabilities and no predictions done for the beam splitters. They don't detect anything. Here, I am saying that we should only discuss things that are actually measured – by observers who must be at least in principle conscious. People look at detectors and what they see may be probabilistically predicted. But if no observer – such as the real human – can perceive the result of anything, then there's no reason to assume that the underlying "events" are objectively real. And it's usually not objectively real.

Haters of quantum mechanics love to paint this dependence of quantum mechanics on observers as some supernatural fantasy. But there's nothing supernatural about it at all. Instead, this statement is a matter of common sense – the only thing we're really saying is that you shouldn't assume that a photon going through a beam splitter has a brain that is just plotting something particular at that moment. Photons have no brains and they're not planning anything while going through beam splitters – which is why physics isn't obliged to described in detail what these photons are just thinking. They're thinking nothing. They don't know whether they're in the first slit or another, whether their distribution in space is described by one wave or another, and so on. And if you constrain yourself to a mental framework in which photons have brains that constantly think about something particular and objectively real, you will become incapable of understanding modern physics just like a creationist who keeps on assuming that God created species manually within a week.

These things are really simple and whoever is rational enough may easily understand them and scream "it's trivial". But some people are just either too stubborn or too stupid or both. To be more general, some people are just way more stubborn than they're intelligent. So they just refuse to learn how to think about these matters correctly. They prefer to think incorrectly and loudly scream that there's nonlocality and/or retrocausality and all this garbage. When you're trying to teach anything important to them, they scream and make sure that your efforts are futile. In fact, they will basically pretend that you haven't said anything at all.

I mentioned some people's inability to listen. Yesterday, among other things, we had conversations with some multiculturalists. Okamura, the Czech-Japanese leader of the top nationalist party, mentioned that since the war, the number of gypsies has increased by a factor of 32 and he views this trend as one of the two main problems facing the country (the other one is mass migration). Well, it's not surprising that this is how a nationalist party views it. I am much less certain that I would classify this demographics as the main problems but it's obviously legitimate that someone may see it in this way – and millions of Czechs almost certainly do.

OK, so a multiculturalist says that it's a crime that he said those things at all. Fine. Non-multiculturalists were obviously not impressed by this attempted "ban". So the multiculturalists added that the factor of 32 is only this high because there has been the Endlösung before that (the final solution, the Gypsy holocaust). So I wrote that I am absolutely against any Endlösung but the fast increase is worrisome despite the pre-history.

Now, a multiculturalist screamed "I am against Endlösung but... – ROTFL". Why was he rolling on the floor? It's simple. Because in their modified grammar, there can't ever be any word "but" following the word "Endlösung". When they say "you're a fascist" or "Endlösung", they assume it's the end of the discussion and they have totally won it. It's the ultimate nuclear weapon they have and they assume that no one dares to say another word after this nuclear weapon is used. Any further debate becomes impossible.

Well, I am fortunately totally immune towards this (not at all nuclear) weapon and my mouth is able to pronounce any word after the word "fascist" or "Endlösung" that I need. When someone tries to shut down the discussion by screaming "fascist" or "Endlösung", the discussion may end for him and other people who are brain-dead but it doesn't end for the people who are reasonable and intellectually honest. And the reasonable people know that by trying to shut down the discussion after the nuclear word, the shutter hasn't won the debate. He has lost it.

The topics are completely different – retrocausality in the delayed choice quantum eraser experiment; arguments on whether or not some demographic changes are worrisome – but the people's lack of will to listen to anything that could contradict what they want to hear and what they're trained to hear is basically the same in all these cases. Too many people are dishonest pompous fools who are simply unwilling to learn anything new and/or impartially check other people's arguments.

And that's the memo.