Apologies for the bad mathematical formatting in this post. I tried to format everything in HTML, rather than my usual habit of using Latex and copying over the images of the equations. The formatting is, unfortunately, even worse than usual doing it this way. Hopefully it is still understandable.
A large part of my work is centred around the interpretation of quantum physics, and in particular my claim that quantum field theory fits well (albeit not perfectly) with Aristotle's metaphysics, in particular the ideas of act and potency, the four causes, and so on. As a consequence of that, I have suggested an interpretation of quantum field theory, which I am attempting to gradually refine and improve (and hopefully learn to understand and express better).
One of the bounds for any interpretation of quantum field theory is that it needs to agree with and explain the physics. In particular, there are various results which are commonly said to exclude various possible interpretations. Two key aspects of quantum physics are interference and entanglement. Interference effects manifest themselves in the double slit experiment, and entanglement in the Einstein-Polanski-Rosen experiment, which is tied to Bell's theorem. I have discussed both of these in detail before.
What I haven't discussed much is another thought experiment which is said to constrain interpretations of quantum physics, which is Wigner's friend. Recently, I have come across two papers which extend the Wigner's friend paper, by postulating a pair of entangled friends. These two papers are available online at the arXiv, 1604.07422 (Frauchiger and Renner) and 1907.05607 (Bong et al.).
In this post, I want to take a quick break from my discussion of Victor Stenger's work, and review these papers, and illustrate how they impact my own philosophy. I will first of all give a discussion of Wigner's friend thought experiment, then Frauchinger's model, and then turn to Bong's model.
Wigner's friend thought experiment can be though of as an extension of Schroedinger's cat. Instead of instead of putting a cat in the box, we put a physicist, and instead of a vial of poison, we have some quantum experiment. So we have two observers, A and F. F is the friend. A is sealed (with his laboratory) in the box. We have a physical system, |ψ〉, which can take two states, say |↑〉 and |↓〉. Initially, this system is prepared in an initial state |→〉 = 1/√ 2 ( |↑〉 + |↓〉). So there is a 50% chance that that it will be measured in a |↑〉 state and a 50% chance that it will be measured in a |↓〉 state. Both A and F know what the initial state is, and what experiment is going to be performed, but F doesn't know when A is going to perform the experiment, and A isn't a very nice friend because he refuses to tell F the result of the experiment. Thus, after A performs the experiment, he records (for example) that |ψ〉 = |↑〉. As far as F is concerned, however the particle is still in a state |ψ〉 = |→〉. So we appear to have a contradiction. How can |ψ〉 be both |↑〉 and |→〉 at the same time?
This apparent contradiction is easily resolved if (as I do, albeit with a bit more complexity than just this) |ψ〉 doesn't really represent the actual state of the particle, but merely our knowledge of it. This is particularly true if we accept (as I do) that the amplitude is always conditional. So if we use the symbol \ to represent conditional on (since the conventional symbol | is part of the notation used to denote a quantum state, and I want to avoid any more confusion than is necessary), I as the known initial state, and M↑ as knowledge that the system was measured in the |↑〉 state, then strictly we should be writing everything as A understanding is represented by |ψ\IM↑〉 = |↑〉, while F's understanding of the system is represented by |ψ\ I〉 = |→〉. There is no discrepancy, because the two friends have a different knowledge of the system. As such, they have different objects on the left hand sides of their corresponding equations.
So Wigner's friend argument might pose a problem for interpretations of quantum physics where ψ represents the actual physical state of the particle (ψ-ontic interpretations). I'm not fully convinced that it does, but it makes life more uncomfortable for them. But for those who use interpretations which allow for ψ to sometimes represent our knowledge of the physical system, then there is no problem.
Frauchiger and Renner's extension
So now things get a bit more interesting (note that my notation is different from that of Frauchiger and Renner, but I am describing the same set-up). We now have two physicists each in their own sealed lab, A and B, and two friends F and G. F is A's friend and G is B's friend. We also have two quantum particles, which I will call a, and b.
Particle b is the same as the one described in the preceding section. I will just need to introduce another bit of notation. The state
|←〉 = 1/√ 2 ( |↑〉 - |↓〉).
So it is the state orthogonal to |→〉
a can be represented by two states, |h〉 and |t〉. I will also define superpositions of these states, firstly the initial state for particle a,
|I〉 = 1/√ 3 |h〉 + √ 2/√ 3|t〉.Particle 3 is going to be sent to F, and he is going to measure it in terms of the basis
|+〉 = 1/√ 2 ( |h〉 + |t〉)
|-〉 = 1/√ 2 ( |h〉 - |t〉).
Particle a, in the state |I〉, is sent into A's lab. He performs his experiment. Two thirds of the time he is going to get the result |t〉 and one third of the time he is going to get the result |h〉. Whatever he does, he sends particle a onto his friend F, who measures it in the basis defined by the states |+〉 and |-〉. But be also prepares particle b, and sends it to the other sealed lab, B. If A measures the result |h〉, then he sends the particle in the state |↓〉. If he measures the result |t〉, then he sends his particle in the state |→〉. So the end result of A's machinations can be summarised by the operator
S = |h〉|↓〉 〈 h | + |t〉|→〉 〈 t |.
Notice that the two particles are now entangled. This is essential to how the experiment is going to progress.
B also performs his measurement, and records the particle in either a state |↑〉 or |↓〉. He sends his result on to his friend G. So to everyone outside the lab B, his actions are represented by the operator
T = (|h〉〈 h | + |t〉〈 t |)((|↑〉〈 ↑ | + |↓〉〈 ↓ |).
A and B, being the worst friends you can imagine, keep the results of their experiments to themselves. But F and G are far more social, and are happy to share their results with each other, after they have both performed their experiments. So now we come to the final measurement. F measures particle a in the |+〉 or |-〉 basis. G measures particle b in the |←〉 and |→〉 basis. There are thus four possible combination of results these two people can get. We can calculate the amplitude for their results using (for example) the formula
A(←-) = 〈-|〈←| T S |I〉.
In particular, a quick calculation shows that A(←-) = 1/(2√3), so one time in 12 F will record result -, and G will record result ←. So far, so good.
But now let's bring B back into the mix, and consider the correlations between F's result and B's result. In particular, we can consider the amplitude that F measures |-〉 and B measures |↓〉. This is given by,
A(↓-) = 〈-|〈↓| T S |I〉 = 0
So what this means is that if F measures the result |-〉, then he can be sure that B measured the result |↑〉. But B can only measure the result |↑〉 if the particles emerged from A in the state |t〉|→〉. But if it emerged from A in that state, then G can never measure it in the state |←〉. Thus the probability of a simultaneous measurement of |-〉 and |←〉 is 0. It can never happen.
But hold on a minute. We have just calculated that the probability is 1/12, i.e. it will happen sometimes. It can't both sometimes happen and never happen. We have a contradiction.
If there is a contradiction, then one of the premises of the argument must be incorrect. Frauchiger and Renner claim that their argument is based on only three premises.
- Premise Q is the statement that all the agents use quantum theory to calculate the probabilities, i.e. that we can express an amplitude using the formula 〈 F| π |I〉, where I is the initial state, F the final state, and π represents the sequence of evolution and measurement operators. The mod square of the amplitude corresponds to the probability. An amplitude or probability of 1 means that we are sure that the event will happen.
- Premise C is the statement that we demand consistency. It basically means that if an agent reasons that some measurement or event is certain (or certain to fail) then another agent using the same theory will come to the same conclusion.
- Premise S is the premise that there can only be single values of a measurement. If an agent establishes that I am certain that a measurement comes out as a particular value, then I must deny that that measurement will not also get a different result. This looks to me to be the principle of non-contradiction (cannot both be X and not X at the same time and in the same sense).
Note, however, that these three premises have technical definitions, which in my view don't fully line up with the one-line summaries I have included at the start of each of those paragraphs.
The claim of Frauchiger and Renner is that one of these assumptions must be violated. The initial version of their paper strongly favoured premise S as being the one that had to go. They saw this as strong evidence for the Everett multiple worlds interpretation of quantum physics. After peer review, they toned down their conclusions, and noted that other interpretations of quantum physics violated either premises C or Q.
Their analysis is that every interpretation of quantum mechanics must violate at least one of these assumptions. The Many worlds interpretation is the only one that violates assumption S, but they are unclear about how it relates to the other two. Otherwise they assign the Copenhagen interpretation, consistent histories and Quantum Bayesian interpretations as violating premise C (consistency), and Bohmian mechanics as violating premise Q (applicably of Quantum Theory).
Their in-depth discussion of Bohmian mechanics is relegated to an appendix, which unfortunately is not really in depth enough for me to verify without spending time to duplicate the calculations they hint at. Their assertion is that Bohm's interpretation just deals with a single universe, so obviously satisfies premise S. I have no problem with that. They then state that Bohmian mechanics applies to the universe as a whole, in that the underlying pilot wave is non-local and everywhere. Since there is only a single object driving the physics, there must be consistency between the observers. Thus Bohmian mechanics must satisfy premise C. They thus conclude that it must violate premise Q. In Bohmian mechanics, agent A, by measuring the state |t〉, cannot conclude that G will measure |→〉. In other words, it is not merely quantum evolution between the two states. They also state that the final result you get (the correlation between F and G's results) depends on the order in which F and G make their measurements, which does not hold in the other interpretations of quantum physics. If true (and I reiterate that I haven't verified the calculation), then this might provide an experimental way of distinguishing between the pilot wave and other interpretations of quantum physics.
Quantum Bayesian interpretations is a subjective interpretation of quantum mechanics, in that quantum states are representations of an agent's knowledge or belief. This is obviously similar to my own interpretation, although not exactly the same once you get down into the details. Here, the problem is working from the conclusion that F's measurement implies that B measures |↑〉, which in turn implies that G will measure |→〉. The issue is in reasoning from B's result to G's result, and the assumption this relies on is C. The authors question whether a weaker form of C might still be allowed to stand.
The Copenhagen interpretation satisfies both Q and S, and therefore is also taken to indicate a violation of assumption C.
Assumption Q assumes that the wavefunction evolution holds until we take the measurement. That means that those interpretations which add to the evolution some form of spontaneous or gravity induced collapse will violate this assumption.
My own interpretation is that the mathematical framework used in quantum physics can be both used to represent the actual state of the particle, and our knowledge of the state of the particle. Obviously, we only use it in one of these senses at a time, but the same formalism is used everywhere. I adopt a particle rather than a field ontology; the underlying quantum fields represent prime matter or a state of pure potentiality, which needs to be actualised by taking on the form of a particular excitation to give an existent substance. Thus the particles or excitations are the objects actually modelled and studies by physics, and the fields don't exist in and of themselves, except by subsisting in the particle. I also distinguish between substance causality and event causality. Substance causality concerns particle creation and annihilation, and asks what particle states did another particle emerge from, or what it could decay into. Event causality discusses questions of which of the various options in practice was chosen. For example, when we take a measurement, then that is a quantum event, because one option out of many is chosen. We also have spontaneous emission and absorption of virtual particles, which are other quantum events. Substance causality is local, and fully determined, in the sense that other physical particles provide a sufficient and complete list of causes. Event causality is indeterminate, in the sense that the physical particles by themselves are not sufficient to explain the event. It is also non-local. This means that if reality is rational (there is a sufficient explanation for everything), then there must be an additional ingredient which is immaterial (otherwise we would be able to include it in the physical representation), transcends space and time (which implies its actions could be non-local), and free in its decisions (in that it could select different events in the same circumstances). This has all the attributes usually assigned to God. In this sense, quantum theory is fundamentally indeterminate. We cannot know what the actual state is between measurements. All we can do is calculate amplitudes for the various paths from the initial to final states, and add them up. All amplitudes are conditional on their premises, which include the initial state but also any measurements of the path which the particle takes. Thus the wavefunction more often expresses our uncertainty. For free particles (which is what we generally consider between measurements), all the spin operators (for example) commute with the Hamiltonian, or energy operator, meaning that there is no reason why one particular basis should be preferred over another. By a measurement (of spin), I mean when a particle becomes a part of a larger system, such that the new effective Hamiltonian prefers one basis over all the others (decoherence). As such, the particle will drop into that basis, and take up either one or the other of the spins. This is a quantum event, so it is indeterminate, and could take either option. The probabilities for each event are (baring the miraculous) determined by Born's rule. Thus when I speak of a measurement, I don't just mean when a physicist is performing an experiment and drawing out a numerical value, but any time when the particle is absorbed into a larger system which modifies the Hamiltonian. So it doesn't rely on there being a conscious observer (although, in the below, I am mainly going to discuss the case where it is a conscious observer).
Entanglement occurs when two different quantum particles are tied together in such a way that events that affect one of them are correlated with events affecting the other. Quantum events come from God's free choice, but we can use the mathematics to predict (probabilistically) how likely it is that God will chose one way or another. In particular God's transcendence implies various symmetries which constrain these amplitudes; and, baring a miracle, leads to correlations between the first measurement on each of the particles. So if you measure an observable on one entangled particle (the quantum event), that tells you something about how God will act (if there is no miracle) when you perform a similar measurement on the other particle. But entanglement only affects the first measurements we perform on the particles. Once we perform a measurement, we is broken, and there is no longer any correlation between any subsequent measurement states.
This is my main criticism of Frauchiger and Renner's no-go theorem, as it stands at least. A sends out entangled particles to B and F. These particles have to be in an entangled state. Sending a particle to B in the state 1/√ 3|↓〉 + √2/√ 3|→〉 makes no sense because the two states are not orthogonal, and therefore this state is not correctly normalised (i.e. the probabilities all add up to one). It is only because of the entanglement with the |h〉 and |t〉 states that Frauchiger and Renner are able to mix up their bases in this way. But as soon as B makes his measurement, the entanglement is broken. The particle is projected into either a state |↑〉 or |↓〉, and there will no longer be any correlation with what is happening at F. Thus There will be a 50% chance that G measures |←〉 and 50% chance that he will measure |→〉 regardless of whatever is happening at F.
So, the calculation goes roughly as this. All amplitudes are conditional on the premises used as the basis of the calculation, so the quantity we are interested in is
There are two possible paths to get this result. Firstly, if B measures a state |↑〉, and secondly if B measures a state |↓〉. G's measurement only depends on the state outputted by B, while the measurements of F and B are entangled, and so ought to be treated together. We can thus decompose the amplitude as
A(←,-\ I) = A(←\↑)A(↑,-\ I) + A(←\↓)A(↓,-\ I).
In terms of the quantum operators and states, this gives,
A(←,-\ I) = 〈 ←|↑〉 〈 ↑|〈 -| S|I〉 + 〈 ←|↓〉 〈 ↓|〈 -| S|I〉.
The numbers are,
〈 ←|↑〉 = 〈 ←|↑〉 = 1/√ 2
〈 ↑|〈 -| S|I〉 = 1/√ 6
〈 ↓|〈 -| S|I〉 = 0
So the final result is A(←,-\ I) = 1/√ 12, just as above, and we get a probability of 1/12.
So this that this interpretation violates Frauchiger and Renner's premise C. But that doesn't mean that it is inconsistent. Every amplitude is conditional on its premises. When they calculated a zero amplitude, they were actually computing something different, namely
A(←,-\ I, ↓) = 〈 ←|↓〉 〈 ↓|〈 -| S|I〉 = 0
But there is no contradiction between the two different amplitudes A(←,-\ I, ↓) and A(←,-\ I) being different. Nor can F draw any conclusions about what G measures. He knows that if he measures a state |-〉, then B would emit the particle in a state |↑〉. But, since this is in a different basis to that measured by G, then G could get either result. Thus F's deduction of what B measures says nothing about the result obtained by G. He cannot draw any conclusions.
Of course, if B did not perform his measurement, then the particles sent to F and G would be entangled, and you would get a different result. This reminds me of the double slit interference experiment. The result changes if you measure which path the particle goes through, for the same reason that happens here.
So I don't think that Frauchiger and Renner's thought experiment actually says very much of interest. If their premise C really did mean consistency, then it would be of the utmost importance. But in reality the two seemingly contradictory results are for two different amplitudes, which becomes clear once we note that all amplitudes are conditional and write in what premises were used to calculate these particular amplitudes.
A more interesting thought experiment would be if A, instead of sending a particle to B which is then passed onto G, emitted a pair of entangled particles, one to B and the other to G. You would then have a situation combining the Wigner's friend `paradox' with an EPR `paradox'. All three particles coming out of A would be entangled together. But that's something to think about, and not something I will explore further here.
The paper by Bong et al does combine a Wigner's friend and entanglement scenario. They use this set-up to derive various inequalities, similar to Bell's inequalities. The violation of these inequalities would imply that one of their premises, which together they call "local friendliness", would be violated.
The set-up is similar to the above. Once again, they have two people in closed laboratories, who I will call A and B, and two friends F and G. This time there is the decay of a particle, leading to two entangled quantum states, one going to A and the other B. The particles are then passed onto F and G, who perform measurements on the individual labs.
We need some notation (note my notation here is different from that of the original paper, to keep things consistent with the above). F and G each have a free choice of N different experiments to perform. For example, these might be measuring the spin of the particles coming out of the laboratories along different axis. These possible different measurements are labelled by the parameters x and y. x=1 represents the case where F just asks A what his result was. Similarly y=1 represents the case where G just asks B what his result was. Otherwise x represent different experiments performed by F and G in ignorance of the results of A and B. A and B, however, always perform the same measurement.
For each measurement, there are assumed to be O possible outcomes. For example, for a spin 1/2 particle (such as an electron), there would be two possible outcomes: spin up, and spin down (+1/2 and -1/2). For a spin 1 particle (such as a photon), there would be three possible outcomes: up, neutral and down (1, 0, and -1). And so on.
We denote a as the measurement result of A, b as the measurement result of B, f as the measurement result of F, and g as the measurement result of G. The final result we are interested in is the probability of getting f and g given x and y, which can be written as
Three assumptions are required to derive the inequalities:
Absoluteness of observed events. An observed event is a real single event, and not relative to anything or anyone. Formally, this is expressed as saying that there is a probability p(fgab\ xy) such that
P(fg\ xy) = Σa,bp(fgab\ xy)where p is such that if x = 1 then f = a, and if y = 1 then g = b. I use a lower case p to denote this probability because it is never actually compared to experiment; it merely characterises the hidden variables of the system. (Bong et al. make this distinction between what they call the empirical and theoretical probabilities, and I am merely following them.)
No super-determinism. Any event is uncorrelated with a freely chosen action at a future time (with respect to a suitable lab reference frame). This implies that a and b are independent of x and y, or
p(ab\xy) = p(ab).
Locality. An event is uncorrelated with any freely chosen action at another location outside the light cone. What this means is that the measurement result f is independent of the measurement choice y, while g is independent of x.
p(f\abxy) = p(f\abx)
p(g\abxy) = p(g\aby)
The super-determinism and absoluteness of observed events assumptions are also required by Bell's theorem. (Perhaps slightly re-formulated to account for the different setups between an EPR and Wigner's friends experiments, but that doesn't make much of a difference to the conclusions.) However, this locality condition is not the same as that used in Bell's theorem, which instead relates to the independence of the measurement outcomes f and g. Bell's theorem requires (for example)
p(f\gxy) = p(f\xy)
This means that this set-up represents a weaker set of assumptions than Bell's theorem (which are called the local hidden variable assumptions). It thus places a more stringent set of conditions on the philosophies. We can add in Bell's version of locality into this system as an additional, and that allows Bong and his collaborators to reproduce Bell's inequality. But they don't use it to get their own inequalities. Their version of locality only requires that the choices made by one experimenter at one location don't affect the results of another experimenter at another location. It does not imply that the experimental results (or the quantum events) at one location are uncorrelated with the experimental results (or the quantum events) at another, as Bell's premise does.
I won't go into the details of how various inequalities are derived from these assumptions, because those details are quite complex (and not, in my view, explained well in the paper). But they derive a family of inequalities which a theory that satisfies these assumptions will satisfy, and which standard quantum mechanics suggests should be violated. They then perform an experiment which sort of reproduces this set-up (A and B are represented by mirrors and interferometers rather than a conscious observer, so an advocate of a theory where consciousness causes wavefunction collapse will complain), and violate some of their inequalities.
So one of the assumptions must be violated. The authors of the paper claim that Quantum Bayesian interpretations and the Multi-world interpretations violate the absoluteness assumption (while satisfying the other two), while Bohmian mechanics violates even this weak form of the locality assumption (while satisfying the other two). I think my own approach also violates the absoluteness assumption, as they have formulated it. I express the internal uncertainties of the system as amplitudes rather than probabilities (only converting to a probability at the last step when we want to compare a prediction for an ensemble of results with a frequency distribution). Taking the modulus square of a sum of amplitudes (what my calculation would require) is not the same as the sum of the modulus square of the amplitudes (what the mathematical form of this assumption requires).
So, not surprisingly, this experimental set-up can't be used to distinguish between the different interpretations of quantum physics. That's not surprising: all the interpretations imply the usual mathematical framework, so one would the local friendliness assumptions to be broken. But it is interesting, again, that the different interpretations break different assumptions. So, for example, if you have independent reasons that that this particular expression of locality is correct, then the experiments violating the inequalities would rule out Bohmian mechanics. If you are certain that this particular expression of the absoluteness assumption is right, then that would rule in Bohmian mechanics.
These sort of studies are, interesting, in one sense, and not in another. The problem with them is that every good interpretation of quantum theory implies the same underlying mathematical framework. (The de-Broglie/Bohm interpretation is possibly an exception here, since it introduces an additional hidden layer to physical reality, which leaves the underlying system deterministic.) They are interesting because the different interpretations might violate different assumptions. Thus if you have independent reasons for supporting one of the assumptions rather than another, then such studies might make a difference. Frauchiger and Renner's claim that their model allows Bohmian mechanics to be experimentally distinguished from other interpretations, is, if true, of considerable interest. But I am a little sceptical of this claim. In quantum mechanics (rather than quantum field theory, where there are challenges in formulating a pilot wave interpretation), Bohm's construction reproduces the usual mathematical framework which then falls on one side or the other of the dilemma. If the orders in which the measurements were carried out makes no difference to the mathematics, then I struggle to see why adopting an underlying pilot wave ontology would make a difference. So I am sceptical about Frauchiger and Renner's result here (albeit given I haven't performed the calculation, so my natural scepticism doesn't count for much).
The first moral to draw from this is to always be cautious when read when someone says that a particular experiment has proved or disproved this particular interpretation of quantum physics. As long as the interpretation (plus maybe some additional assumptions or experimental data) correctly implies the mathematical framework of the standard model of particle physics, then no experiment is going to disprove it, unless that experiment first disproves the standard model (which would be very big news). Disproving the standard model doesn't disprove the philosophies, which tend to be more general than that and support a range of possible models. For example, if there were three Higgs Bosons out there rather than just one, that would mean that the standard model would need to be modified, but it wouldn't make much of a difference to the underlying philosophy. But you can't experimentally disprove the philosophies without first disproving the theory. Nor can an experiment distinguish between different interpretations or philosophies which lead to the same mathematical theory. Frauchiger and Renner's original version of their paper was claimed to have proved the multi-worlds interpretation. It was never going to do that, as their revised version acknowledged.
The second moral is to always be careful about what such studies claim. Their premises are usually represented in two different ways. The first is in words, which ties to the philosophy, and the second in symbols, which ties to the physics. These two ways of expressing the premise don't always coincide. It is the mathematical expression which is more important in the derivation of the result. An example of this is Bong's absoluteness of observed events assumption, which sounds daunting, until you into the mathematical detail and release that it is merely a statement about how probabilities are constructed from other probabilities.
If we are to distinguish between the different interpretations, it has to be on the basis of wider philosophical consistency, and the explanatory power of the interpretations. Studies such as this are useful in helping us understand more precisely the assumptions behind each model. And that gives us a clearer vision of how they do, or don't, fit in with a particular philosophical outlook. But that is pretty much the only use such studies have.
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