# Problem regarding quantum entanglement and special relativity

So this question is about the strange phenomenon of quantum entanglement, or "spooky action at a distance", as Einstein called it. In particular, it's about the simplest conceivable case of quantum entanglement that you'll probably ever see, namely the one with two particles that are entangled in such a way that if one particle is observed to have spin up, then the other will have spin down, and vice versa.

So the story usually goes something like this: imagine that you have two particles that first interact in such a way that their combined spin wave function becomes a superposition of one having spin up and the other spin down, and vice versa. Then separate the two particles by some arbitrary distance. Now, if you measure the spin of particle A, say, and it turned out to be spin up, then simultaneously the wave function of the other particle (which I'll call B) will collapse and yield spin down. At first, I thought this was a problem: according to special relativity, simultaneity is relative, and so how can you talk about two events (the collapses of the wave functions of the two particles) being simultaneous in any, objective, meaningful way?

So what I did, was that I asked this exact question to professor Brian Greene in one of the live streams in his "Your Daily Equation"-series on YouTube. Here's a link to the video: https://www.youtube.com/watch?v=iGDNzW59E3M (at 39:18 in the video, he starts answering my question). Now, he said that the two collapses don't have to be simultaneous. It is just that we can choose a reference frame in which they are. But of course, if we instead go to another reference frame in uniform velocity compared to the other, then the collapses won't be simultaneous anymore, according to the rules of special relativity. But he did point out that since we can always choose some reference frame where the two events happen at the same time, the two events must be spacelike separated, and so no physical signal can ever reach particle B in time. Thus, the "spooky action at a distance"-business still applies.

But to me, there seems to be another problem too. For if we are in a frame where the two events are not simultaneous, then it will take some time for the wave function of particle B to collapse after the measurement of A, which we'll assume yielded spin up. So what if I then observe B before the collapse? Then, there is still a non-zero probability that B will yield spin up, and so that might actually happen. In other words, we might have that both particles yield spin up, which was supposed to be impossible. What am I missing here?

• Why do you say that "it will take some time for the wave function of particle B to collapse after the measurement of A"? Jun 29, 2020 at 15:24
• This question has already been answered, and I know that my reasoning is wrong, but my problem was that it seemed like Brian said that there could be a time difference between the collapses of the wave functions of A and B in some particular reference frames, in which case it could happen that there would "take some time for the wave function of particle B to collapse after the measurement of A". Jun 29, 2020 at 16:00

You are asking the right kind of question, and the main thing I want to say is to warn you that a huge amount of stuff has been written on this and only about $$0.00001$$ percent of all that stuff is worth reading.

These questions go to the heart of what we mean by a 'state' when talking about quantum systems. Most people try to frame their discussion by invoking the notion of a quantum state and writing down state vectors (kets in Dirac notation). This mostly sets one up to fail because one has adopted from the start the very language which leads to confusion when dealing with correlations of the type considered in Bell inequalities.

One can avoid some of the muddles by refusing to do that, and instead adopt an approach with more of a quantum field theory flavour. That is, one makes the calculation like an input--output problem, or a type of scattering process. One does not make statements about collapse of wave function, one simply calculates the quantum amplitude for whatever overall process from start to finish one wishes to know about.

The main thing I would want to say is that in the entangled scenario neither party has an individual state in and of itself, for the physical quantity under discussion (e.g. direction of spin angular momentum). One must train oneself to stop thinking of it as if it did---as if the spin was "partly up and partly down" or something like that, or could "collapse".

In order to unpack that final comment a little further, consider the following sequence of events:

Sequence A:

1. entangled pair is created at some source O, and sent to places A and B---one part to A, the other part to B
2. an operation is applied at A
3. the pair is brought together at C where it is manipulated and measured.

Now consider the following sequence of events:

Sequence B:

1. entangled pair is created at some source O, and sent to places A and B---one part to A, the other part to B
2. an operation is applied at B
3. the pair is brought together at C where it is manipulated and measured.

I have in mind an operation that did something---it was not an identity operation. For example, consider an operation such as a rotation which had the effect that the final state, observed at C, is orthogonal to the state prepared at O.

Now the important thing about an entangled state is that sequence A and sequence B can have the same overall effect. That is, the change caused by the operation in the middle of the sequence is the same change. It causes the same result at C in the two cases. So we have two operations which have one and the same effect. This means that, at least as far as their effect on this entangled pair is concerned, they are one and the same operation, just carried out different ways, like if you turn a coin over using your left hand or your right hand, the effect on the coin is the same. But the thing being compared to a coin here is at two different locations---the spacelike-separated locations A and B.

It is mistake to say that in sequence A the operation adjusted some physical property of the particle at A. It did not, because the same result can be obtained by doing nothing at all at A and instead following sequence B. So one must say that the operation, whether at A or B, adjusted a physical property of the pair, a property which cannot be assigned to either of A or B on their own.

The kind of property we are talking about here could be the direction of the spin angular momentum, and the associated magnetic field in the case of a magnetic dipole, or it could be polarization for a photon or internal energy for an atom. Those are all experimentally realised examples. But in principle it could be anything at all.

Finally, instead of the sending the particles back together at the end as in my two sequences, one could instead have them interact with some other parties such as measuring devices and then bring the results of those interactions together at C. This is the scenario usually described in experiments of this kind. The above, with suitable modifications, still applies.

I will finish by doing something I have deliberately resisted doing. This is to write down an entangled state in Dirac notation. As I already warned, to do that is already to embark on a misleading way of thinking, because it suggests that each part of the system has its own state. But that is the very thing that is not true. The Dirac notation here is $$| \psi \rangle = \frac{1}{\sqrt{2}} \left( |\downarrow \rangle_A |\downarrow \rangle_B + |\uparrow \rangle_A |\uparrow \rangle_B \right)$$ or some other entangled state. It gives the impression that the particle at A has a physical property called direction of spin, and so does the particle at B, and the two are correlated. But I claim this is not so. To see why, apply a rotation to the particle at A. The state becomes $$| \psi_1 \rangle = \frac{1}{\sqrt{2}} \left( |\uparrow \rangle_A |\downarrow \rangle_B + |\downarrow \rangle_A |\uparrow \rangle_B \right).$$ So the spin at A has changed, right? Well no. If we return to $$| \psi \rangle$$ and apply a rotation at B then we get $$| \psi_2 \rangle = \frac{1}{\sqrt{2}} \left( |\downarrow \rangle_A |\uparrow \rangle_B + |\uparrow \rangle_A |\downarrow \rangle_B \right)$$ and the crucial thing here is that $$| \psi_1 \rangle = | \psi_2 \rangle.$$

It is the attempt to put into words what is going on here that fills so many pages of writing.

The essence of the calculation by John Bell, and which is also on view in the above, is, I think, that we must abandon the way of thinking in which every physical property is physically represented or carried as a property of an individual thing separated off in its own little spacelike region. The physical world simply is not like that. It is like that in most of science, but not in this little corner of quantum physics---a corner that proves to be not so little because it is involved in things like phase transitions and chemical reactions, and possibly in processes significant to biology such as the way enzymes work.

On this view, the result of a process such as a measurement at A is not best described by saying that it immediately caused a change at B. Rather, one would say that the spin direction and associated things such as the local magnetic field is here physically carried by a two-particle entity that is present at A and B and that cannot be correctly understood as two individual things. It can be understood as two things in some of its properties such as mass, but not for entangled properties such as spin direction in this example. In short, one has a single physical embodiment, not two, and other parts of the physical world can interact with this physical embodiment at either A or B or both. The physical property of spin direction is here physically embodied in a non-local way.

• Could you please clarify what you mean when you say "neither party has an individual state in and of itself"? I seem to not understand what you are referring to here. Jun 23, 2020 at 7:43
• @FelisSuper Thanks for asking; I realised I was being a bit terse so I extended my answer quite a bit. Jun 23, 2020 at 10:14
• Wow, this was quite interesting. The one thing I still don't understand is how Brian's ideas fit into this. Where in the math does it say that you can have a time-delay after a measurement on the system, where it seems to take time for the whole wave function to collapse? If the two particles constitute one and only one system, then surely the measurement should always immidiately affect the whole system. If not, then it seems like it distinguishes one particle from the other. Jun 23, 2020 at 12:21
• Also, I would just like to know, is your answer concerned with a special interpretation of quantum mechanics other than the usual Copenhagen interpretation? Charles' answer seems to be concerned with relational quantum mechanics, which sounds interesting, but surely the Copenhagen interpretation must have something to say too. Jun 23, 2020 at 12:34
• @FelisSuper I haven't watched the video but I gather from your text that Brian Greene adopts the language of states and collapse. I would say that such language is not entirely wrong but it is somewhat unhelpful; it is not giving the right intuitions. Nothing at all happens at A when an operation at B takes place, but the pair does acquire connections and correlations to other things at B. Jun 23, 2020 at 13:23

This arises from a fundamental misconception in the early days of quantum mechanics, that the quantum state describes the physical state of a particle. In fact the quantum state describes an observer's knowledge of the particle (well illustrated in Schrodinger's cat and Wigner's friend).

So, if Alice measures particle A, she acquires knowledge of particle B, causing Alice's wave function for particle B to collapse. This has no physical impact on particle B, and Bob, who measures particle B, cannot detect Alice's result, or even whether Alice has done a measurement. Alice's wave function for particle B collapsed, but Bob's did not.

Only later, when Alice and Bob bring their results together, do they detect a correlation. It turns out (Bell's theorem) that there is no classical explanation for the correlation they detect. But in any case, there is no violation of relativity.

• Comments are not for extended discussion; this conversation has been moved to chat.
– Chris
Jun 29, 2020 at 15:22
• This answer gives the misleading impression that there is now a consensus in the physics community around the epistemic interpretation of quantum mechanics. That's a perfectly valid interpretation, but not the only one. In fact, I would probably argue that with the increase in popularity of the many-worlds interpretation over the past 30 years, if anything the pendulum of popular opinion has probably (slightly) swung back in the direction of the ontic interpretation. Jun 30, 2020 at 1:50
• @tparker, science is not a matter of opinions or consensus, otherwise religion would be science. It is about evidence and strict logical (mathematical) argument. The answer says nothing concerning consensus, but only about the mathematical structure of quantum mechanics. Jun 30, 2020 at 9:00
• @CharlesFrancis Your answer claims that a quantum state is only a state of knowledge and does not describe the physical state of the particle, and that claim is not in any way implied by the postulates of quantum mechanics. This perspective is only one of several perspectives that all fully reproduce the predictions of quantum mechanics. Jun 30, 2020 at 12:13
• @tparker, Not so. This is explicit in the Born rule, and also clear in en.wikipedia.org/wiki/Dirac%E2%80%93von_Neumann_axioms. You talk of hand wavy arguments like many worlds, which are incapable of correctly reproducing mathematical structure. I suggest a deep study of mathematical foundations. Jun 30, 2020 at 17:47

Make your measurement at B. You'll get either "up" or "down". Repeat the whole experiment many times. You'll get "up" about half the time.

Is this because the wave function is collapsing as you measure, or because it's collapsing shortly before you measure? There's no way to tell. Either description leads to the observed 50/50 probability.

Observers who prefer one frame will describe things one way. Observers who prefer a different frame will describe things a different way. They'll all make the same predictions about anything we can observe, which is all that matters.

• Comments are not for extended discussion; this conversation has been moved to chat. Jun 23, 2020 at 13:33

imagine that you have two particles that first interact in such a way that their combined spin wave function becomes a superposition of one having spin up and the other spin down, and vice versa.

1. Spin and magnetic dipole of subatomic particle are correlated.

2. The production of superposition means that these two magnetic dipoles are oriented anti-parallel for example.

3. The spatial orientation of the whole system is unknown. Furthermore this implies that our knowledge of which particles North-South-orientation is more up or down to an arbitrary direction is zero.

Then separate the two particles by some arbitrary distance. Now, if you measure the spin of particle A, say, and it turned out to be spin up, then simultaneously the wave function of the other particle (which I'll call B) will collapse and yield spin down.

You have to ask yourself, does the particles state changes if you measure it? And indeed, it does. Because, not knowing the direction of the magnetic dipole (or spin), you use a polarizing grid and in some cases the particle goes through and in some not (due to point 3 above). The same for the second particle. You get a correlation in the case that both particles successful overcome the barriers. In the other cases the knowledge still is zero.

What collapses is our knowledge about the orientation of the particles.

What is clear is that we set up experiments and proofed many times that our setup produces correlated particles (by spin = magnetic moment). The experiment is state of the art. We know empirical the outcome of the experiment. And now one tell you that it is magic to get the correlated results?

As you was told in the other answers, think and don’t believe.

then it will take some time for the wave function of particle B to collapse after the measurement of A

This is your problem. It doesn't take any time- wave function collapse is instantaneous, in every reference frame. That is to say, once a measurement has been taken, all further measurements are taken on the collapsed wavefunction. There is no violation of relativity because "the wavefunction has collapsed" is not something that is physically observable, it's just an interpretation of events that can be made by an observer after the fact.

It is tempting to claim that the collapse must take some amount of time- surely it must be constrained by the speed of light, at least? But there is no causal relationship here. The physics of quantum mechanics only gives correlations among measurements taken on entangled particles, regardless of which measurement is taken first. You can interpret these correlations as measurement causing the wavefunction to collapse, but you shouldn't expect this interpretation to play nice with the your preconceptions of special relativity.

• Wow, it's funny how you just happened to post this answer when I got online :). But anyways, I think I know the problem in my reasoning. As you can see, there was a lot of activity here a few days ago, and I learned a lot from the others. What you just said is one of the things I learned from them. But thanks for your answer anyways :). Jun 29, 2020 at 15:15

the concept of what happens before loses meaning when the events have a space-like separation. In fact geometrically speaking, this whole business of saying before for such events is ill-defined. There is no strict time ordering possible between both events precisely because any ordering is observer-dependent

What do they really mean when you say that in some frame, one event happens before the other? they mean that you are using your current moving frame to establish a time slice and use that time slice to induce an ordering by convention (think of it as your personal standard of cosmic timekeeping). This time slice of now extends to spatial infinity and it is mathematically defined as all events that could have been reached if you had sent a light signal to that event $$-T$$ seconds ago, and you will receive back a signal response from that event $$T$$ seconds in the future from now. If you change from moving frame, this time slice will be tilted toward the direction of movement in 4D geometry

• Comments are not for extended discussion; this conversation has been moved to chat.
– Chris
Jun 29, 2020 at 15:22