## Bell’s inequality, entanglement and the demise of local reality – I

The best way to make sure you understand something, is to try and explain it to someone else.  Zeilinger’s discussion of these matters in “The Dance of the Photons” ( see https://luysii.wordpress.com/2010/11/23/book-review/) is marvelous.  You don’t have to know any quantum mechanics, any more math than counting and simple inequalities, and very little physics (all of which he explains fairly clearly, but over pages 58 – 142, 271 – 288).  If you have time, read Dr. Z rather than this.  If not, get out your polaroid sunglasses join Alice and Bob, two freshman physics students.

Each of them gets a box with a red and a green LED (light emitting diode) and a computer to record the flashes the LEDs make and the time the flashes occur.  The boxes are on either side of the Danube (Zeilinger is from Vienna after all) and between them in a sewage treatment facility under the Danube is a black box, hooked by separate fiberoptic cables to Alice’s and Bob’s boxes.  That’s it.  Then the data (red and green flashes) comes in and they try to figure what lies underneath. Each box has a switch with 3 settings — zero, plus and minus.

To shorten the narrative  up a bit, they find that the black box in the center is pumping out photons (particles of light).  When they hit Alice or Bob’s box they make either the red or the green LED flash.  The LEDs on each box flash irregularly about once a second (never both at once).  Each run of 200 flashes (which their computers record if they want) is about half red  and half green (rarely exactly half) — this is where the fact that Zeilinger is an experimental physicist as well as a theoretician shows up.

What does a photon ‘look like’?  You can think of photons like the jumping jacks you played with as a kid.  There are 6 arms at right angles to each other.  One pair of opposite arms is where the magnetic field is, another pair of opposite arms is where the electric field is, and the third pair is the direction the photon is moving.  Note that the fields are always in a plane at right angles to the direction of motion.  It’s possible to imagine the 4 arms of the jack (the electric and magnetic  fields)  rotating around the direction of motion, but in this set up that doesn’t happen.

Light from a bulb is rather different. For one thing it is made of zillions of photons.    In addition, all possible orientations of the 4 arms of the jack are present among them.  This is where your polaroid glasses come in.  They’ll transmit light in only one particular orientation of the electric and magnetic field without blocking it.  Call this orientation zero.  (see figure #13 of Zeilinger p. 68).  Every photon getting through the polaroid  is oriented this way (call it vertically polarized).  This is why the light getting through is called a polarized beam.   A second polarizer in the path of the light will transmit these vertically polarized photons perfectly (if it is oriented the same way as the first) or not at all (if it is oriented at right angles — 90 degrees to the first).  Just knock your polaroids out of your frames, put them one after the other in front of a bulb and rotate the second and see for yourselves.

So all light gets through the second polaroid if it is oriented the same way as the first, and none gets through if it is oriented at right angles.  What happens with orientations of the second polaroid between 0 and 90 degrees? Interestingly, the answer was found out over 2 centuries ago by Malus.  The amount of vertically polarized light getting through the second polarizer  varies as the square of the cosine of the angle between the two. Don’t panic — all you have to know is that with a 30 degree difference from the first polarizer t 75% of the vertically polarized light gets through and with a 60 degree difference 25% of the light gets through. At 45 degrees half gets through.

The one thing Zeilinger doesn’t explain terribly well is the polarized beam splitter (PBS) and how it works.  Suffice it to say that it takes a polarized beam of photons and splits it, so that the photons getting through go one way, and those not getting through go another, so both types can be counted by a detector.  It’s possible to  rotate the PBS to any angle you want (see p. 283).  It you rotate the PBS to 30 degrees one direction will register 75% of the photons and the other will register 25%.    This is where the red and green LEDs come in.  Each is connected to a separate detector on the other side of the PBS.  Now remember,  the light going in to the PBS is oriented in just s single direction (because it’s polarized).  This is where the randomness of quantum mechanics comes in.  There is simply no way to tell what an individual photon arriving at the PBS will do (and they are hitting the PBS one at a time).  Quantum mechanics does give an accurate prediction of what large numbers of them will do (it turns out to be exactly the same as Malus’s law).

So this explains the 3 settings on Alice and Bob’s boxes.  Minus is rotation of the PBS 30 degrees to the left, zero is no rotation and plus is rotation 30 degrees to the right.

Now (at last) for some data. The first thing Alice and Bob notice is that half the time the red LED lights up, half the time the green.  Never both, and it doesn’t matter how they set the 3 switches (plus, minus, zero).  So they think that this is boring and are rather depressed.  They look again at their data (which includes which LED lit up, the time it lit up and the setting of the three switches).  When both switches were set to zero, about 20% of the time both boxes registered a photon at exactly the same time (to within 1 billionth of a second).  When this happened, if Bob’s green LED flashed so did Alice’s). Regardless of which LED flashed, the other box flashed the same color LED.   This happened when both boxes had their settings the same (both minus, or both zero ….).

Great !   They didn’t care about the 80% of the time that photons arrived at the two detectors at different times — they put this down to experimental noise, imperfections in the fiberoptics, or the source, or the detector. Zeilinger the experimentalist again.

So what to make of this.  The simplest (and wrong) explanation is that the black box under the Danube emits two photons at once, each polarized the same way (vertical or horizontal).  Why is this wrong?  You have all the information you need.  Think about it a bit before reading the next paragraph.  One of the felicities of Zeilinger’s writing is that he lets Alice and Bob stew about this a bit (after they’re told by the grad student running the experiment the their explanation is wrong) before cluing the reader in 5 pages later.

Here it is.  If both switches are set to minus, the black box emits photons polarized horizontally or vertically to the PBS oriented 30 degrees to the left.  If both switches are set to zero, the black box emits photons polarized horizontally or vertically relative to it.  If both switches are set to plus …   So how does the black box know the correct polarization of the photon pair it is emitting?   One explanation is that it in some way ‘knows’ the settings of the PBSs and emits photons appropriately.  This won’t work, as it’s possible to switch the orientation of the polarizers while the photons are in flight and the results are the same.  Iimagine just how tricky this experiment was to set up — Louisa Gilder describes these tribulations in her book “Entanglement”.

Lets assume that the black box can emit photons in just the three orientations of the PBSs (minus, plus, zero).  What happens if the photon pair emitted in the zero orientation hits the PBS at minus 30 degrees.  We know.  75% of the photons will go one way and 25% the other — they will do this randomly, so that there will not be a perfect correlation between the color of the LEDs on Alice and Bob’s boxes for these photons.

Somehow when one photon goes one way in Alice’s PBS, its mate goes the same way in Bob’s.  This experiment has been done with longer and longer separations between the boxes (now up to several miles, and perhaps more), and the results are the same.   Ladies and gentleman, you have just met entanglement.  This is what Einstein called ‘spooky action at a distance’.  Even better, Zeilinger has shown you the experimental evidence for it.

One way to look at the result is that the black box emits a pair of photons with no particular orientation, until they hit the PBS and are forced to make a choice (which somehow is always the same for both).  This is the essential strangeness of the quantum world (not quantum mechanics which describes it).  Remember, the data always comes first in science.

Well, suppose each photon came with instructions about how to behave for each of the three orientations of the PBSs?  These are the celebrated hidden variables, and Bell showed that they can’t possibly explain the results you’ve just seen.  But that’s for next time.