In 1935, Einstein, Podolsky, and Rosen wrote a paper which laid out very clearly one of the great mysteries of quantum mechanics, namely that there are intrinsically nonlocal correlations in a system with many degrees of freedom. This nonlocality was described by Einstein as "spooky interaction at a distance." From a classical standpoint it is indeed strange, but there have been many experiments which show that it is definitely real. In this lecture we will try to understand what the mystery is, and how experiments by Alain Aspect have confirmed it.

Observables for two-level systems

We have seen a few two-level systems in this course: The ammonia molecule, and the kaon (the latter was in homework). From an abstract point of view, all two-level systems are the same, and we can enumerate all physical operators which can be measured. These are the Hermitian operators. A Hermitian operator has real eigenvalues, and orthogonal eigenvectors. If you measure a physical quantity, you had better get a real number. The states corresponding to two different values of an observable had better be orthogonal. With a finite number of states, an operator is just a matrix. A Hermitian matrix $\cal O$ is one that satisfies ${\cal O}^\dagger={\cal O}$. The most generic two-by-two Hermitian matrix is
\begin{eqnarray}
{\cal O}&=&\left(\begin{array}{cc}
a+z&x-i y\\
x+i y&a-z
\end{array}
\right)\\
&=& a {\cal 1}+ x \sigma_x + y \sigma_y + z \sigma_z,
\end{eqnarray}
which defines the Pauli matrices $\sigma_x,\sigma_y,\sigma_z$. The labels come from the fact that one can make a vector from the operators ${\vec \sigma}=(\sigma_x,\sigma_y,\sigma_z)$. The Feynman Lectures on Physics, Vol 3, chapter 6 has an extended argument about why this can be thought of as a vector. The basic reason is that electrons have a property called spin, which is associated with its intrinsic angular momentum of magnitude $\hbar/2$. One typically uses a basis where
\begin{equation}
|\uparrow\rangle = \left(\begin{array}{c}1\\0\end{array}\right)
\end{equation}
represents the spin pointing in the $\hat z$ direction. In that case the
the matrices $S_x=(\hbar/2)\sigma_x,S_y=(\hbar/2)\sigma_y,S_z=(\hbar/2)\sigma_z$ measure the angular momentum along each of these directions. These different matrices do not commute -- meaning that measuring in different orders gives different results.

Polarization of light is another good two-level system, so much of the logic of polarizers can help you with your intuition here. "Measuring" is sending a photon at a polarizer, and asking if it goes through or not. In fact, the EPR experiment can be done with spin-1/2 particles, or with photons. You are all familiar with the fact that if you stack polarizers the order matters.

The cleanest version of the EPR experiment requires three different measurement, corresponding to measuring the spin along the $\hat z$ axis, and $120$° to either side. The operators are
\begin{eqnarray}
S_z&=&\frac{1}{2}\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\\
S_{120}&=&\frac{1}{4}\left(\begin{array}{cc}-1&\sqrt{3}\\\sqrt{3}&1\end{array}\right).\\
S_{-120}&=&\frac{1}{4}\left(\begin{array}{cc}-1&-\sqrt{3}\\-\sqrt{3}&1\end{array}\right).
\end{eqnarray}
All three of these have
eigenvalues $1/2$ and $-1/2$, so in any measurement you will get one of these.

The eigenstates of these operators are
\begin{eqnarray}
|\uparrow\rangle &=& \left(\begin{array}{c}1\\0\end{array}\right)\\
|\downarrow\rangle &=& \left(\begin{array}{c}0\\1\end{array}\right)\\
|\uparrow 120^{\circ} \rangle&=&\left(\begin{array}{c}1/2\\\sqrt{3}/2\end{array}\right)\\
|\downarrow 120^{\circ} \rangle&=&\left(\begin{array}{c}\sqrt{3}/2\\-1/2 \end{array}\right)\\
|\uparrow -120^{\circ} \rangle&=&\left(\begin{array}{c}1/2\\-\sqrt{3}/2\end{array}\right)\\
|\downarrow -120^{\circ} \rangle&=&\left(\begin{array}{c}\sqrt{3}/2\\1/2 \end{array}\right).
\end{eqnarray}
At this point it is convenient to also introduce column vectors
\begin{eqnarray}
\langle\uparrow| &=& \left(\begin{array}{cc}1&0\end{array}\right)\\
\langle\downarrow| &=& \left(\begin{array}{cc}0&1\end{array}\right)\\
\langle\uparrow 120^{\circ} |&=&\left(\begin{array}{cc}1/2&\sqrt{3}/2\end{array}\right)\\
\langle\downarrow 120^{\circ} |&=&\left(\begin{array}{cc}\sqrt{3}/2&-1/2 \end{array}\right)\\
\langle\uparrow -120^{\circ} |&=&\left(\begin{array}{cc}1/2&-\sqrt{3}/2\end{array}\right)\\
\langle\downarrow -120^{\circ} \rangle|&=&\left(\begin{array}{cc}\sqrt{3}/2&1/2 \end{array}\right).
\end{eqnarray}
Traditionally one writes the inner product as
\begin{equation}
\langle\uparrow| \uparrow 120^{\circ} \rangle = \left(\begin{array}{cc}1&0\end{array}\right) \left(\begin{array}{c}1/2\\\sqrt{3}/2\end{array}\right) = \frac{1}{2}
\end{equation}

Here is a question which tells you if you understand what all this means. Suppose I prepare a spin in the $|\uparrow\rangle$ state. I then measure it along the $120^{\circ} $ axis. What is the probability that I get up?

The answer is that I calculate the overlap with the eigenstates of the measurement operator, and square it. So in the present case there is a 25\% chance that I will measure $\uparrow$, and a 75\% chance that I will measure $\downarrow$.

The EPR thought experiment

Imagine a device which produces two photons, which travel in opposite direction, such that the spins are always pointing in the same direction, but the direction the spins point in is undetermined. We can actually build these devices (for example using spontaneous parametric down-conversions), and the spin-state of the photons can be written
\begin{equation}
|\psi\rangle=\frac{|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle}{\sqrt{2}}.
\end{equation}
Now you might object that the $\hat z$ direction seems special here. It turns out it is not. For example, if you rotate this $90^{\circ} $,
\begin{eqnarray}
\uparrow&\to& (\uparrow+\downarrow)/2\\
\downarrow&\to& (\uparrow-\downarrow)/2
\end{eqnarray}
so ignoring all of the multiplicative factors
\begin{eqnarray}
|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle&\to&
|(\uparrow+\downarrow)(\uparrow+\downarrow)\rangle+|(\uparrow-\downarrow)(\uparrow-\downarrow)\rangle\\
&=&|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle
\end{eqnarray}
All the cross-terms vanish, and it is unchanged by the rotation.

Einstein, Podolsky, and Rosen pointed out a strange feature of this state. Suppose I measure the first photon, and determine that it is spin up. A subsequent measurement of the second photon is then guaranteed to be spin up. This is true, even if there was no time for any signal propagating from the first measurement point to reach the second point. Thus, in the Copenhagen interpretation, where the first measurement "collapsed" the wavefunction, the information about the collapse travels faster than the speed of light.

EPR did not like this fact, and argued that to preserve "local realism" the two photons had to already have known that they were both spin-up when they were emitted. Bell showed that EPR's idea was wrong. There is no local realism in quantum mechanics. He derived a set of inequalities which any Einstein-style local theory must obey. Later, Alain Aspect did the experiment, and showed that the real world violates those inequalities.

Bell's inequalities

Bell's argument was that for Einstein to be right, the photons must not only know what direction they should be pointing in along the $\hat z$ direction, but also in the $\pm 120^{\circ} $ directions. He imagined an experiment where the two detectors are each randomly set to independently measure one of these three directions. For every set of photons they are randomly set to a new direction. [In Aspect's experiment they are even set after the photons are produced, so that there is no way the photons could have known.]

Suppose, as Einstein suggested, the photons had a code-book. It might say something like: If the detectors is in the $\hat z$ direction I will be up, if it is in the $+120^{\circ} $ direction I am down, and if it is in the $-120^{\circ} $ direction I am up. Both photons have the same codebook, so that if the two detectors are in the same direction you will get a coincidence (which is the quantum mechanical rule for this state). There must be an ensemble of these codebooks if we are to reproduce quantum mechanics. Let $P(\sigma_1,\sigma_2,\sigma_3)$ be the probability that a code is produced with $\sigma_j$ in direction $j$. We must have
\begin{equation}\label{in1}
C=\sum_{\sigma_3}P(\uparrow\uparrow\sigma_3)=P(\uparrow\uparrow\downarrow)+P(\uparrow\uparrow\uparrow)=1/8.
\end{equation}
There is a 50\% chance that $\sigma_1=\uparrow$, and then quantum mechanically there is a $1/4$ chance that a measurement in the $120^{\circ} $ direction will be the same as in the $\hat z$ direction. Similarly
\begin{equation}
\sum_{\sigma_3}P(\uparrow\downarrow\sigma_3)=P(\uparrow\downarrow\downarrow)+P(\uparrow\downarrow\uparrow)=3/8.
\end{equation}
But by symmetry $P(\uparrow\downarrow\downarrow)=P(\uparrow\downarrow\uparrow)$ and we conclude
\begin{equation}
P(\uparrow\downarrow\downarrow)=P(\uparrow\downarrow\uparrow)=P(\uparrow\uparrow\downarrow)=3/16.
\end{equation}
Substituting this into Eq.~(\ref{in1}) we find
\begin{equation}
P(\uparrow\uparrow\uparrow)=-1/16.
\end{equation}
This is not good. Probabilities cannot be negative.

Thus either the hidden variable theory is wrong, or the quantum mechanical probabilities are wrong. Bell essentially gave this same argument, but codified it in a convenient inequality, putting a bound on how small the quantity $C$ in Eq.~(\ref{in1}) can be if you have local realism. At the time it was a shocking result, as it was assumed that the question of "local realism" was more philosophical than physical.