Quantum Entanglement and non-Locality

d.w.rowlands [at] gmail.com

The following is part of the material I covered while helping substitute-teach a two-hour class for high school students as part of the MIT Educational Studies Program's Splash 2011.

The teacher became seriously ill the day before the class was to be taught and he didn't leave a syllabus, only the course title "Faster-Than-Light Travel", so a physics grad student and I had twenty-four hours to come up with one from scratch. He ended up teaching an hour-long lecture on Special Relativity, while I put together an hour-long lecture on various things that are or look like ways to get around it. One part of this was a discussion of quantum entanglement that I wrote with a lot of input from a physicist friend. Writing this was hard, but I feel like I learned a good deal from it, because I never previously understood how we knew that hidden variables couldn't exist. I know I read a number of Scientific American articles on this topic when I was in elementary school, but I didn't really understand them, and this issue wasn't really covered in the quantum classes I did take in that much detail. Anyway, what follows is a modified version of my lecture notes, which I thought might interest people.

Quantum Entanglement

It's commonly known that Einstein's theories of relativity restrict communication to the speed of light. Now it's time to talk about something that Einstein himself was never comfortable with: the ways in which quantum mechanics may involve instantaneous transmission of the correlations between particles. It turns out that quantum mechanics may require this, and yet it does so in a way that doesn't allow any information to travel faster than light, so causality is preserved.

Einstein-Podolsky-Rosen Paradox

One thing that quantum mechanics predicts is that it is possible to "entangle" two particles, so that their properties are related in a known way without knowing what these properties are. For example, electrons have a property called "spin", which you can think of as essentially a vector pointing in some direction in space. You can produce a pair of electrons that you know have opposite spins, without knowing the spin of either electron.

Now, for a single electron, quantum mechanics says there is a limit to how much we can know about its spin. We can, for example, measure whether the x-component of the electron's spin is positive or negative, but in doing so we destroy its initial state, so we can never measure the y-component of its initial spin: it now has a fifty-fifty chance of having either value, regardless of what its initial state was.

What if we have a pair of entangled electrons that we know have opposite spins, and we take them far enough apart that we can measure each one's spin before a signal traveling at the speed of light could get to the other one? We might measure one's spin along the x-axis and one's spin along the y-axis, and then know the x- and y- components of each electron's spin.

Yet quantum mechanics says that once we've measured the x-component of one electron's spin, we've perturbed the system, and we shouldn't be able to measure the y-component of either electron's spin: this seems to indicate that measuring the first electron's x-component sends an instantaneous message to the other electron to enter a perturbed state where its original y-component can't be measured.

Einstein objected that this was absurd, as it involved instantaneous transmission of information, which relativity forbids. He argued that this meant that the electrons must have hidden variables carried with them locally that allowed their measurements to be truly independent, so that each one could be measured locally, without caring about events happening far away.

Bell's Theorem

Einstein proposed the EPR paradox in 1935, and spent the last twenty years of his life searching in vain for an improved theory that would re place quantum mechanics and resolve the EPR paradox without requiring measurements to be non-local.

In 1964, nine years after Einstein's death, John Steward Bell published a paper that showed that any theory of the sort Einstein wanted–any theory that allowed each electron in the previously-discussed pair to be measured locally without caring about the other electron–would make experimental predictions that contradict the predictions made by quantum mechanics. In other words, for Einstein to be right, quantum mechanics can't just be incomplete, it has to be wrong. This is particularly useful because the experiment Bell proposed can actually be done.

Suppose we have a large collection of pairs of entangled electrons, and we divide it in half, giving one electron in each entangled pair to Alice and one electron in each entangled pair to Bob. Now, Bob stays on Earth and Alice travels to Proxima Centauri with her electrons. On a predetermined day, they each select an axis at random to measure their electrons' spin with respect to. Alice then travels back to Earth so she can compare her measurement results with Bob's. If they both picked the same axis, they'll obviously get consistent results in every case: if Alice measures a spin-up electron, Bob will measure a spin-down electron and vice versa. If they pick perpendicular axes, there will be no correlation between their measurements: half of the electrons Alice measured as spin-up, Bob will measure as spin-down, and the other half he'll measure as spin-up. Where things get complicated is if the axes they pick aren't perpendicular or parallel.

In this case, quantum mechanics predicts that the correlation will vary as cos(θ), where θ is the angle between the axes of the two detectors. What Bell showed is that any theory where local, real variables fully describe the spins of the electrons, the correlation will instead vary linearly with θ, as you would expect it to classically. (A nice graph of this, from Wikipedia.)

So we now have two separate predictions that contradict each other: all we need to do is an experiment that lets us see which actually describes reality. We can't send Alice to Proxima Centuri, but we can send her far enough away that, with atomic clocks and fast enough measurements, she and Bob can randomly pick axes and measure the spins of their electrons before a speed-of-light signal could pass between their labs. A number of such experiments have been done, and all of them have given results that agree with quantum mechanics and disagree with Einstein.

So the correlations of quantum states can travel instantaneously, despite the speed of light.

Causality Preserved

All is not lost, though. As you'll recall from the discussion of special relativity, there are two fundamental problems with faster-than light travel. One is that an object with mass traveling at the speed of light would have to have infinite energy. There's no reason to think that this correlation is transmitted by something with mass, though, so that's not an issue.

The bigger and more fundamental problem is that if information is ever transmitted superluminally, causality and the nature of time itself break down. Fortunately, it turns out that we can prove that this non-local correlation doesn't actually allow any information to be transmitted faster than light. This is called the no-communication theorem.

Imagine the set-up we discussed before. Alice on Proxima Centauri and Bob on Earth have a set of entangled electrons that they're going to measure the spins of. Each of them picks an axis to measure spin along at random. Later on Alice will travel back to Earth and they can compare their results and their choices of axis. Once they get to Earth, they'll find that Alice's data set–call it A–and Bob's– call it B–have a correlation of cos(θ) and conclude that their electrons were communicating instantaneously somehow.

But suppose, instead, that Alice wants to try to send a message home from Proxima Centauri instantaneously this way. She can't control what spins the electrons have when she measures them: she can only vary two things. Whether she takes a measurement at all, and what axis she uses to do so. Can Bob figure these things out by measuring his own set of electrons?

• Suppose he measures his electrons "first", before she has. In this case, no matter what axis he picks, he should have a fifty-fifty chance of getting either a positive or a negative spin.
• Suppose she measures her electrons "first", along the same axis he later measures his along. This could be achieved if they agree what axis he'll use, so her message is encoded in her choice of axis. Then, each of his measurements is guaranteed to come out the same as hers, but since her measurements came out randomly with a fifty-fifty chance of each result (and since he doesn't know what her results were until she comes back to Earth to tell him), he still sees a each electron having a fifty-fifty chance of having each spin.
• Suppose she measures her electrons "first", along an axis perpendicular to the one he used. In this case, the spins he measured have no correlation with the spins she measured, and have a fifty-fifty chance of having any result.
• Suppose she measures her electrons "first", along an axis that differs from his by some angle that is neither zero nor ninety degrees. If she measures her first electron to have positive spin, then he has a 50% + δ chance of measuring his first electron with a positive spin, and a 50% - δ chance of measuring his first electron with a negative spin. But she has a fifty-fifty chance of getting a positive spin for her first electron, and no way of controlling what spin she actually gets. So his measurement has a 50% * (50% + δ) + 50% * (50% - δ) chance of coming out spin-up, where δ is a function of the angle she chose. However, this expression evaluates to 50% + (50% * δ) - (50% * δ) = 50% chance of coming out spin-up, regardless of the value of δ.

Thus, in this situation it is impossible for Bob to figure out whether Alice has measured her electrons' spin, or what axis she measured them along, until she comes back to Earth to tell him. Furthermore, even once they meet again to discuss their results, neither one of them can prove that their measurements happened "first": from a relativistic point of view, they were simultaneous.