Hello, everybody. Today's lecture is about Bell's Experiment. This experiment was
designed by the physicist John Bell in the 1960s, and this experiment has provided
deep insights into the nature of quantum mechanics, into the nature of
entanglement. And, even today, it's an, it's a topic of active research in quantum
cryptography, quantum information, quantum computation. Okay, so, so, let's start
with a philosophical question. Can we experimentally test whether we live in a
quantum world? What do we mean by this? Well, all the quantum weirdness, the fact
that measurement outcomes are probabilistic, that just looking at the
system changes it's state. The fact that particles do not have trajectories, and
instead, they are in this weird kind of super position where they take all
possible paths simultaneously at some amplitude. Is all this an intrinsic
property of nature or does it reflect the fact that we only have an incomplete
picture, that, that quantum mechanics is an incomplete theory? That is, if you knew
more about the mechanism behind the scenes, if you knew more about the gears
and, and pulleys that are acting behind the scenes, would it actually look much
more natural? This is what Einstein believed. Okay, so the kind of mechanism
that Einstein was looking for, that acts behind the scenes, was a local and
deterministic mechanism. By local, what that means is, that there is no action at
a distance. Of course, local also means that there's no faster than light
communication an the theory of relativity holds. And deterministic, meaning that
there are no, no such probabilistic outcomes to measurements that the system
actually has definite values for physical quantities like, momentum or speed, or
spin, etc., or, or, or bit or sine, etc. I'm sure you've heard the quote,
Einsteins's quote, God does not play dice with the universe. Okay, so this is what,
what we are looking for and in the previous videos we talked about the EPR
experiment, the Einstein, Pedalski, Rosen thought experiment, which was designed to
precisely think about these issues. Now in the 1960's, in 1964, John Bell designed a
remarkable experiment. This was an honest to goodness experiment, not just a thought
experiment. And, it was an experiment which had one of two outcomes. So, if you
do this experiment, and the outcome is outcome one, we'll see what all these
outcomes mean later. But if, if, if the first outcome happened , then you could
conclude that nature is inconsistent with quantum mechanics. That quantum mechanics
is false. But that nature can be explained by some kind of local hidden variable
theory. It's consistent with some local hidden variable theory. On the other hand,
if the second outcome holds, then this would imply that nature is consistent with
quantum mechanics, but it's inconsistent with any local hidden variable theory.
Meaning, that if the second outcome were true, then the kind of theory that
Einstein had spent the last twenty years of his life searching for, could not
possibly hold. It would have been experimentally proved that, in fact all
such theories are ruled out by the way nature behaves, can be observed to behave.
Now, the Bell experiment relies on sophisticated properties of entangled
cubits. Actually of, of the, of the Bell state. And the Bell experiment has been
performed numerous times. And the results have always have been consistent with
quantum mechanics, and inconsistent with local hidden variable theories. So, always
it's outcome two within, within experimental error that's, that's held.
Okay, let's look at Bell's experimental setup So, Bell assumed that the apparatus
for this experiment is divided into two parts which are far enough apart, that
during the course of the experiment, light does not have enough time to travel from
one to the other. So, the two pieces are completely isolated from each other. Of
course, this can be achieved by making the, the duration of the experiment so
short, that you could actually have these two parts of the apparatus at two ends of
your, your lab. Okay, so now, for all purposes, let's think of these two parts
of the apparatus as two boxes. The first box gets us input. One of two, you know
bit values 01, which we'll call X. And the second box, gets us input a bit, which
we'll call Y. What the boxes are supposed to do is, they are supposed to output also
a bit, zero or one. And what they are trying to do is, if both the input bits, X
and Y are one, then the output bits, A and B are required to be different from each
other. So, they could be either 01 or ten. In all the other three cases, we want the
outputs of the two boxes to be the same. Either 00 or eleven. Now, what Bell was
able to show is that, if the boxes, if the physics of the boxes is described by local
hidden variable, local hidden variable theory, then they cannot succeed with, in
this task with probability exceeding three quarters or .75. So, no matter what the
physics of these boxes is, no matter what the physical theory is, as long as it's
local, it's a local hidden variable theory, the chance of success is at most
three quarters. On the other hand, he was able to show that, if the two boxes share
a bell state, meaning that there's one cubit in the first box, and a cubit in the
second box, so, that these two cubits are entangled, and they form this bell state
00 plus eleven. Then this, there are certain measurements that you can make on
these two cubits, that the two cubits can make, in such a way that they succeed in
this task with probability as high as .85. And so, if you perform such an experiment,
if you actually realize this experiment and if within your error tolerances, you,
you showed that your success probability in this task is strictly greater than .75.
Then you'd have shown that nature is inconsistent with any local hidden
variable theory, and it's consistent with quantum mechanics. Actually, to show that
it's consistent with quantum mechanics, you'd have to show that, that you're
success probability to within your error of tolerance is, is really .85. Okay. So,
let's, let's now see, why a local hidden variabl e theory does not allow you to
succeed in this task with probability greater than three quarters. So, the fact
that its a local hidden variable theory, and the fact that these two boxes cannot
communicate with each other, what this implies is that output of the first box
can depend on X, but it cannot depend on Y, the input to the second box. And
similarly, the output of the second box cannot depend upon X, the input of the
first box. Also, the fact that we want to show that the success probability cannot
be greater than three quarters. For contradiction, let's assume that the
success probability is greater than three quarters. Well, in this case, the boxes
cannot afford to be incorrect on any of the four possible inputs 00, 01, etc, etc.
Okay. So, now suppose, so, what this mean is that, if the input was X=0 and Y=0,
then the two output bits must be the same. A must be equal to B. So, let's assume
without, without loss of generality that, A = B = zero in this case. Now, by
locality, if we were to switch only one of the inputs, let's say Y and switch Y, make
Y=1 while leaving X=0. So, now the first box will still output A=0. But since the
two boxes must output the same value on this particular input, X=0, Y=1. So, in
fact, the second box must also output B=0. We can use the same reasoning if, instead
we leave the input Y=0 and now switch X=1. So, we, we now reason that, well, of
course, B still must be zero, but now, since A must be equal to B, in fact, in
this case, also A=B=0. Okay. Now, let's argue that this further implies that when
we gave the two boxes, the inputs X=1, Y=1, they still must output A=0 and B=0.
But this is a contradiction, because in this case, they were supposed to output
different bits. So, they were supposed to output either A=0 an B=1 or A=1 and B=0
So, that's a contradiction. So, let's see how this last step works. This is a little
subtle. Okay. So, you see, in the case X=0, Y=1, the outputs were, the first box
output zero, and the second box output zero. Well, so, let's pay attention to the
second box in this case. Well, from the second box's point of view, it was getting
us input Y=1. It doesn't know whether the first box got input X=0 or X=1. So, it
must get the same output regardless of which input it, the first box got. But in
the case that the first box got input zero, the second box output B=0. So, it
must still output B=0 in the case that X=Y=1. Symmetrically, we can also argue
that from the first box's point of view, when it was given input one, it must
output A=0, because it doesn't know whether it's in the case X=1 and Y=0 or
whether it's in the case X=1, Y=1. So, we have argued that in the case X=Y=1, both
boxes must still output zero and so, they are both outputting the same bit, which is
a contradiction. Okay. So, so we've proved our first assertion here that if the boxes
are described by local hidden variables, they succeed with probability at most
three quarters. The second assertion that if the boxes actually follow quantum
mechanics, if they are allowed to share a bell state, then they can succeed with a
probability as high as .85. This is actually somewhat more involved, and to do
this we need to understand some properties of bell states. Which we'll do in the next