I’m currently working on a book about relativity, but I still spend a fair amount of time thinking about quantum issues. A lot of this won’t make it into the book, because I can’t assume people will have read How to Teach Physics to Your Dog before reading whatever the relativity book’s title ends up being, and because explaining the quantum background would take too much space. But then, that’s what I have a blog for…

Anyway, the section I was working on yesterday concerned causality and faster-than-light travel, specifically the fact that they don’t play well together. Given Tuesday’s Toddler Toy Teleportation post, it was inevitable that I would start thinking about EPR-type entanglement experiments and how you would deal with them in the context of relativity. So, here’s the scenario: imagine we have a pair of quantum particles– electrons that can be in either spin-up or spin-down states, say– and we prepare them in an entangled state such that when they are measured they will always have the same state. If you look for spin-up vs. spin-down, they will either be both up or both down, never one up and one down. If you look in for some combination of states– spin-up plus spin-down vs. spin-up minus spin-down, say– you will always find both in the same state, whatever that may be.

Quantum mechanics says that this correlation will *always* exist, no matter what state you look for, and no matter how far apart the two states are when you measure them. This effect seems to fly in the face of relativity, and led Einstein to derisively call it “spooky action at a distance.” Unfortunately for Einstein, this prediction has also been comprehensively confirmed experimentally– the correlation between entangled states really does exist, and it does not seem to be limited by the speed of light.

Of course, this becomes really weird when you look at this in the context of relativity, which got me distracted for a little while yesterday afternoon. I think I more or less understand it now, and having spent some time trying to get my head around it, I figured I might as well type it up for the blog. So let’s imagine a scenario where we split our two particles up, and give one to a dog sitting at rest in a laboratory, and the other to a cat who flies off in a UFO at half the speed of light. Both dog and cat wait some time before measuring the state of their spins, and then much later get back together to compare their results. We can represent this scenario in a diagram that looks like this:

This is what’s called a spacetime diagram, and as you might guess from the clever name, plots what happens to the dog and the cat in both space and time. The distance they move in space is represented along the horizontal axis, while the time that passes is represented along the vertical axis. The diagram is scaled so that light, shows as the two red dashed lines, follows a line that is 45 degrees from the vertical, moving one foot to the right or left for every nanosecond of time moved upward, into the future.

The dog’s motion is represented by the brownish vertical line– she doesn’t move at all in space, but marches relentlessly into the future at a rate of one nanosecond per nanosecond. The cat’s motion is the black line, which moves in both space and time, one foot to the right for every two nanoseconds upward into the future. They start out at the same place at the same time, at the origin of the axes drawn here, then the cat moves off while the dog stays put.

According to the dog, she measures her spin at the point in space and time marked by the “1,” while the cat’s measurement is a short time later and some distance to the right, at the point marked by the “2.” The horizontal dashed lines represent particular instants in time, according to the dog, so we can clearly see that the dog makes her measurement, and then the cat makes his about half a nanosecond later.

The dog, then, would say that her measurement determined the outcome of both spin measurements. If she measured spin-up, that instantly and absolutely fixed the cat’s spin in the spin-up state, so the cat’s subsequent measurement only confirmed the already determined state of the spin. If she measured spin-down, the cat would inevitably obtain spin-down.

OK so far? Here’s where it gets weird: relativity tells us that the dog and the cat disagree about the passage of time. Specifically, the dog looking at the cat’s clock thinks that the cat’s clock is running slow, while the cat looking at the dog’s clock thinks the dog’s clock is running slow. More importantly, they also disagree about the synchronization of clocks– if the dog prepares a whole bunch of clocks at different positions so they all show the same time, the cat will say that the clocks are out of synch, with more distant clocks set a bit ahead or behind, depending on the direction of motion. And if the cat prepared a similar set of clocks at different positions moving along with the cat at the same speed, the dog would say that clocks at different positions showed different times.

We can show this on our diagram by doing basically the same thing we did with the dog, and drawing a bunch of lines indicating single instants in time according to the cat. The resulting lines look like this:

Points representing a given time, according to the cat, fall on slanting lines in the diagram. This has a very interesting consequence for our entangled state measurement. The two measurement events are plotted at the same points on the graph, but as you can see by counting lines or just looking at the numbers, the two measurements take place in the opposite order according to the cat. The cat would say that he measured the state of his spin first, and then the dog measured hers half a nanosecond later.

From the cat’s point of view, his measurement is the one that determines the state of both spins, while the dog’s just confirms the result. They both get spin-up, says the cat, because *his* spin-up result instantaneously put the dog’s spin into the spin-up state.

This would seem to be a huge problem. They can’t both be right, so whose measurement is it that does the job? The really weird thing is that it just doesn’t matter.

Why doesn’t it matter? Because no information has been passed from dog to cat, or from cat to dog. The only way they know about the correlation between their results is when they get back together later on and compare results (we can imagine that they do this many times, with many different entangled pairs, so they can determine the probabilities of all the possible outcomes. At that time, the pattern becomes clear, but until they compare lists of results, all they have is a random string of spin-up and spin-down results (or whatever other measurements they want to look for.

There isn’t a clear causal relationship between the two measurements, because there doesn’t have to be. And, really, it would be problematic if there were a relationship, because of what we see from the diagrams. If the dog’s view that her measurement determined the cat’s result were somehow the correct one, then the cat is in the odd position of seeing the effect (his measurement) take place before the cause (her measurement). If the cat’s view was the correct one, then the dog would have a problem with causality. Either way, it would be bad for physics– if you end up with a theory where effect can precede cause, it’s really difficult to construct any kind of coherent model of the universe.

“So, okay,” you might be saying, “the question of whose measurement caused the correlation is moot, because the correlation was there from the start. Both measurements were always going to come out spin-up, no matter what order they were made in, so it’s not surprising that they see a correlation.” This is a nice idea, and it’s more or less what Einstein was hoping for when he wrote about this kind of scenario with Boris Podolsky and Nathan Rosen in 1935. This would be a “local hidden variable” theory, in which the outcomes of the measurements are predetermined but unknown to the dog and the cat (the “hidden variable” part), and do not depend in any way on what the other animal does or when he or she does it (the “local” part, because the measurement is determined only by factors in the immediate vicinity of the measurement).

It’s a nice and comforting idea. It’s also dead wrong. In the 1960′s, the Irish physicist John Bell thought carefully about this problem, and proved a mathematical theorem showing that the correlations predicted by quantum mechanics can not possibly be satisfied by a local hidden variable theory. That is, there are experiments you can do whose results cannot possibly be explained by a model in which the results of the measurements were determined in advance. The result of the cat’s measurement depends on what the dog measured, and vice versa.

People have done these experiments– first John Clauser and colleagues in the 1970′s, followed by a really beautiful series of experiments by Alain Aspect in the early 80′s, and numerous others in the intervening decades. These experiments show conclusively that the quantum version of events is correct, and local hidden variables can’t be the real explanation (and there are other experiments with things like GHZ states that show the same sort of thing– local realism is pretty well dead). Nobody has done this yet with one of the measurements being made by somebody moving at half the speed of light (I’d love to see the grant proposal for that one…), but there’s no reason to expect that the result would be any different (in fact, you can argue that the existing experiments already cover this, given that there must be some set of observers moving at high speeds who would disagree about the order of Aspect’s measurements).

Which leaves us with a really strange situation. The results of each measurement depend on the other measurement, but the order in which those measurements are made is different according to the two different observers. Somehow, they both get the same result (or results in accordance with Bell’s theorem, if they’re not both measuring the same thing), even though the order in which the measurements are made is different for the two observers.

How do you resolve this? That’s why this isn’t going in the relativity book: because I don’t have a good answer. It just happens to work out that way, because that’s how the universe works. If you have further questions, all I can say is– LOOK! THE WINGED VICTORY OF SAMOTHRACE! (*scampers off*).

The one thing you can fall back on is that it doesn’t really matter *how* the measurements get correlated, because there’s no way to use it to send a signal from one observer to the other. As long as the only effect of entanglement is a correlation between two lists of random numbers, there’s no problem for causality or relativity, just for human physicists trying to fit some kind of coherent narrative to the whole thing.