PH301
MODERN PHYSICS
FALL 1999
EINSEIN-PODOLSKI-ROSEN PARADOX
EPR Paradox
Two electrons, one axis: The Pauli exclusion principle requires that no two electrons in an atom occupy the
same quantum state, so if they occupy the same spatial state, they must have different spin states, such as
spin-up and spin-down. More commonly, though, the two electrons will both be in an uncertain state:
50%-50% spin-up and spin-down. Here comes the problem Einstein had with this idea: As soon as I
measure the spin of one electron to be, say, spin-down, instantly the other electron is no longer 50%-50%
spin-up and spin-down but is now 100% spin-up. A measurement on an electron over here seems to have
an instantaneous (faster-than-light) effect on the other electron over there. Recall from discussions in
relativity why signals faster than light are a problem (order of cause and effect).
Two boxes, one ball in each box: Let’s look at an analogous classical situation. Suppose we have boxes
instead of electrons, and in analogy to spin either up or down we have the box contain either a black ball or
a white one. The classical analogue for uncertainty is that we just don’t know, so we seal the boxes, so the
experimenters don’t know the colors. We send one box off to Siberia and the other to Antarctica. At
precisely ordained times, experimenters at each place open the box to see the color of the ball (analogous to
measuring the spins). We repeat the procedure many times to be sure we see any patterns.
The results of the experiment are that each destination finds what appears to be a random assortment of
black and white balls, each being equally likely. We also find that whenever the box in Siberia is found to
contain a black ball, the box in Antarctica is found to contain a white ball, and vice versa. Does this make
us conclude that finding a black ball in Siberia causes the ball in Antarctica to turn white?! No, we assume
that the balls had those colors all along.
So far the results of quantum uncertainty and classical uncertainty are similar. We need a somewhat more
complicated situation to notice the differences.
Two electrons, different axes: Suppose once again that we have two electrons, which are known to be
opposite in spin, but their individuals spins are uncertain. Once again, if experimenters measure the
vertical components of both spins, they will never be the same. But now let our experimenters be free to
choose a different axis in which to measure the spin component. These other axes may be chosen in such a
way that if the vertical spin of one electron and the new-axis component of the other are measured, they
will sometimes be the same (either both +, or both -), let’s say 75% of the time. On the other hand if both
electrons’ spins are measured along the new axes, the measurements will be the same only 25% of the time.
Classical analogue
Two boxes, two balls in each box: Let’s try to make a classical analogue. We now have two properties of
the box we could measure, so let’s divide the box in half, call one half side A the other half side B. Each
half contains a ball (the color of one ball would correspond to the vertical component of spin, the color of
the other ball to the new-axis component). Once again, we send the boxes off to distant parts, where the
experimenter can choose which side to open, and we repeat the procedure many times looking for patterns.
Note: since measurement of spin along one axis destroys all information about spin along the other axis
(just like measuring position disturbs momentum and vice versa), we will have a specially rigged box. If
one side is opened the ball contained in the other side is destroyed, and there will be no way to determine
what color it was.
The results of the experiment are that once again each destination finds what appears to be a random
assortment of black and white balls, each being equally likely. The statistics on the correlation of colors
are shown in the table below. The rows correspond to which half of the box the experimenter in Siberia
opened; the columns correspond to which half of the box the experimenter in Antarctica opened. In the
intersection of the row and column is the percent of the times that the colors are the same at the two
locations (For example black in Siberia and black in Antarctica).
side A was opened in Siberia
side B was opened in Siberia
side A was opened in Antarctica
AS and AA same 0% of the time
BS and AA same 75% of the time
side B was opened in Antarctica
AS and BA same 75% of the time
BS and BA same 25% of the time
Q2: Suppose you are once again in charge of putting balls in boxes and sending them off to their
destinations. Could you set up a way to fill the boxes that would match the experimental results? How
would you do it? (Ponder this a little, but don’t waste too long, because it can’t be done.)
To see why it can’t be done, consider the following questions:
Q3: Given that AS and BA are to be the same 75% of the time, and AS and AA are never to be the same.
What fraction of the time should AA and BA be the same, that is, what fraction of the time should you put the
same color balls in both halves of the box sent to Antarctica?
To help answer the question, consider the circle below. The top half of the circle represents the half of the
time that As is black, and the bottom half represents the time As white.
As black
As white
Q4: Given the fraction of the time should AA and BA are the same (your answer to Q3) and the fact that B A
and BS are the same 25% of the time, what are the limits on the fraction of the time AA and BS could be the
same color?
Q5: Is the value in the table within this range?
The answer to Q5 is no. There is no way these quantum statistics can come out of a classical example.
Quantum results contradict classical statistical reasoning. How, then, can we explain them?
Least of Three “Evils”: The technical jargon for the assumptions that are made in classical physics is
“separable local realism.” We have three assumptions (realism, locality, and separability), any (or all) of
which may be the incorrect assumption. Below we explore parting with each assumption:
A1: Realism. Since only one box is opened at each place, and the color of the ball in the other half can
never be determined, should we assign any reality to the color of the other ball? If the color of the other
ball has no reality, then there is no sense to asking, “What fraction of the time should AA and BA be the
same color?” Side A and side B will never both be opened in Antarctica, so colors AA and BA can never
both be known. Only one color (i.e. only one spin component) has reality in a given run. The above line of
reasoning stops at Q3, and no contradiction arises.
This is the standard interpretation of quantum mechanics. It is in this interpretation that the uncertainty is
not just a matter of not knowing, but a matter where there is no reality to the value of spin component (or
any other quantity) when there is uncertainty or spread in spin component values.
If one wants to insist on the reality of quantities like all components of spin, there are two more
possibilities  are they any better?
A2: Locality (meaning no signals travel faster than c). If the boxes were equipped with faster-than-light
communication and internal spray-paint, then the opening of the first box could instantly signal the other
box to spray paint the balls the right colors so when it is opened the colors will match the statistics. There
is a rival theory of quantum mechanics that is like this. Each electron has definite real spin components,
but the electrons are in instantaneous “communication” via a guide wave. This one is tempting until you
remember relativity results. If the first box opening causes the spray-painting in the other box, and the
boxes are far enough apart ( x2  x1  c(t 2  t1 ) ), then there will be reference frames in which the spraypainting happens before the “first” box is opened to cause it!
A3: Separability. Perhaps the two measurements are not actually distinct, but are inseparable. This is
related to the idea that if you knew ahead of time which side of each box were going to be opened, then you
would have no trouble setting up the ball colors to match the statistics (try it), and there would be no
contradiction. Suppose it were predetermined which component of the spin would be measured. If this is
the case, though, why does the experimenter seem to have free will to determine at the last moment how the
experiment will be done and which component will be measured?