Monday, December 5, 2011

Entanglement and the Crossed Polarizers

I said last week that there was something wrong with the whole narrative concerning entanglement, and today I’m going to explain it. It has to do with the central role of Bell’s theorem in the ongoing debate. What I realized only a year ago is that Bell’s theorem hardly matters. The barn door was already open and the horse long gone before Bell came up with the business of the 22.5 degrees.

Don’t get me wrong. What Bell did was extremely clever, and he showed how to close an important philosphical loophole in the argument of local realism. But what people don’t seem to realize is that for all practical purposes, local realism was already in a shambles before anyone thought of varying the angle of the crossed polarizers.
I wrote last week about how Einstein pointed out the philosophical problem with two particles shooting off from each other with opposite momenta. According to the theory, the actual trajectory of either particle was completely indeterminate up to the surface of a sphere, until the moment when one of them was detected. At that moment, the wave function of the second particle collapsed simultaneoulsy: the measurement of one had affected the properties of the other.
This was the clear and unmistakeable implication of the theory, and it should have been extremely troubling. However there was a catch: no conceivable experiment could distinguish this philosophical nightmare from the more prosaic explanation that the particles had simply been endowed with their complementary momenta at the moment of separation; that the randomness in their detection was merely a lack of information on our part. To be sure, the theory was clear on the distinction: but that distinction remained, so it seems, only theoretical.

People seem to think that this happy state of affairs came crashing down at the moment Bell proposed his experiment with the crossed polarizers in 1964. That’s what I don’t understand. For me the disaster occurred in 1950 when Bohm proposed the experiment with spin states. The disaster didn’t depend on varying the angle of the polarizers: it should have been evident from the get-go.
We know how the experiment must look if two electrons are created with opposite spins: let one be in the positive z direction and the other in the negative z direction. If we set up two Stern-Gerlach detectors at opposite ends of the lab, we know what must happen. When one detects an “up” electron, the other must detect a “down” electron. There is nothing mysterious about this.

(Yes, I know the Stern Gerlach apparatus does not work on charged electrons but only on neutral atoms: but the theory is the same and for all I know, modern experimenters are able to adapt Stern Gerlach to charged species.)
Where the problem occurs is when we get a stream of electrons at each end of the lab, randomly up and down: what happens is each time detector A clicks “up”, detector B clicks simultaneously “down”. That’s a real problem.

But how is this different from the first situation, I hear you ask, where I said there was nothing mysterious? Sometimes the particle detected at A is up, so B must be down. And vice versa. Each detection stream appears random, but compare the streams and you get perfect anti-correlation. It’s all very ordinary, isn’t it?
No it isn’t! It is indeed possible to prepare pairs of electrons in complementary states, one up and one down (or at least it’s possible to write down an expression for the wave function!) and it’s very clear what must happen when we detect them: if we detect one of them at A with spin up, the other must be detected spin down at B…with a 75% probability! This is the result we get if the particles are endowed with opposite spins at the moment of creation: the detection streams are anticorrelated but not to the extent of 100%. It is only a 75% correlation.

Why is it only 75%? Because in practise, there is no source which prepares electrons in states of alignment only along the z axis: any real source must produce pairs of electrons anti-aligned along a random axis. So, for example, if they are aligned along the x axis, the z polarizer at A will detect its electron up or down with 50% probability; likewise, and completeley independently, the polarizer at B. You can see there is a 25% chance that both polarizers will detect up coincidences, and another 25% chance they will detect both down coincidences. You simply cannot get perfect anti-correlation with this kind of setup.
Unless, that is, you prepare the electrons in a very different type of state. It’s the quantum state we’ve alluded to already whereby the two electrons have opposite spins but there is no actual axis along which the spins are defined, until the moment of detection. That’s the mysterious entangled state that leads to all the philosophical headaches about local realism, and you don’t need to tilt your polarizers to 22.5 degrees to come face-to-face with it. It was, or should have been, obvious from the moment Bohm wrote his spin-modified version of EPR in 1950. There was no need to wait for Bell to come along in 1964 to set off all the excitement.

What I don’t understand about the whole history is why I don’t read anywhere about experiments to detect the perfect anti-correlations predicted by Bohm in 1950? Why does everyone only talk about the crossed-polarizer results motivated by Bell’s Theorem? To be sure, the later experimenters record their data at a variety of different angles, so the Bohm correlations are part of the record. But why does nobody ever talk about them as being significant?

2 comments:

Anonymous said...

This is an excellent explanation that is difficult to find as clearly anywhere else.

Marty Green said...

Thanks, anonymous. I find the level of discourse in the world of physics to be very distressing. It's a bunch of know-it-alls repeating what someone else has already said a hundred times, and if you try to inject a different perspective into the discussion, they just call you a crackpot.