Okay. So, in this video we'll finally get
to the, to Bell's experiment. Okay. So,
let's, let me review where, what we're
trying to do in Bell's experiment.
Remember that in 1964, Bell devised this
remarkable experiment, which had one of
two outcomes. If outcome one is true, the
nature is inconsistent with quantum
mechanics, and in fact, can be explained
by some local hidden variable theory. But
if outcome two is true, then, nature is
inconsistent with any local hidden
variable theory. It rules out every local
hidden variable theory, and in fact,
nature is consistent with quantum
mechanics. And so, Einstein's dream of
finding this more complete picture for
nature, you know, more complete theory, is
doomed. Okay. So, and remember this was
the abstract setup that we have. Apparatus
was divided into two pieces which were at
distant ends of the lab, so, that during
the course of the experiment, light did
not have enough time to travel from one to
the other. And then, we give to each of
these, so, each of these pieces of the
apparatus we called boxes, and we gave
each one, we input a bit 01 and each of
these spit out, or output a bit. And what
we wanted was, if both of the input bits
are one, then we want the output bits from
the boxes to be different. But in all the
other three cases, we want the outputs to
be the same. We also show in the first
video, that if the boxes are described by
local hidden variables, then they cannot
succeed with probability greater than
three quarters in this task. We also
claimed that Bell showed that there's a
way to use, in quantum mechanics, using
entanglement to actually succeed at this
task with probability as high as .85. So,
in this video, we are going to see how
exactly this is achieved quantumly. Okay.
So, the way it's achieved is by using
entanglement. So, the two boxes, each of
the two boxes is going to contain a cubit
and these two, two cubits are going to be
entangled with each other, in this bell
state, 00 plus eleven. Now, when the input
is given, what we are going to do, i s we
are going to measure the first cubit in
one or another basis. So, we are going to
measure it in, in some basis which we'll
pick. Okay. So, we'll measure it in, in
some particular basis, you, you perp,
which depends upon, which basis depends
upon whether the input was zero or one.
Similarly, we are going to measure the
second cubit in the second box, in some
bases V, V perp, depending again, on
whether the input Y is zero or one, which
pieces we choose depends upon the input.
And then, if the, if the outcome of the
first measurement is, U, will, the box
will output zero. And if it's U perp, it
will output one. And similarly, if the
outcome of the second measurement is V,
the second box will output zero. And
otherwise, it will output one. Now, in the
last video we talked about what's the
chance that the two outputs match? And we
said, the chance of that is exactly given
by this. So, you look at the angle between
U and V and if that angle is theta, the
chance that you have matching outputs,
that your output zero in both cases are
one and both cases is exactly. So, the
probability of matching outputs is exactly
cosine squared theta. Okay, so what we
want to do, is we want to set things up
cleverly, so that, in the three cases
where we want the outputs to match, cosine
square theta is close to one, and in the
cases that the two matches, two outputs
are supposed to not match, they are
supposed to be different, we want to make
sure that cosine square theta is almost
zero. Right? Or sine square theta is close
to one. Okay, so how do we achieve this?
So, we have to pick these angles
carefully. And in fact, it's a, once you
think about it a little bit, there's sort
of a unique way of picking these angles to
maximize these probabilities. That some of
these probabilities and that's, that's
done like this. So, let me color code it,
because I have to, I have to use several
possibilities. So, okay. So, lets, lets
draw the axis, the measurement basis for
the first box first. So, in the case that,
that X=0, let's say that w e'll do the
measurement in the U knot, U knot perp
basis, which is just the standard basis.
And in the case X=1, we just rotate this
basis by 45 degrees or Pi by four. And
let's call this space as U1, and U1 perp.
Okay. So, now what about the second box?
How do the measurement? So, if Y=1, we do
the measurement at an angle of Pi by
eight. So let me, let me draw these dotted
lines for the coordinate axis and then, we
have an angle of Pi by eight. And let's
call this V knot, V knot perp. On the
other hand, if the, if Y=1, then we do
measurement at minus Pi by eight, right?
So, sorry. That minus Pi by eight, and
that's our basis. P1, P1 perp. Okay. So,
now what do we want from all these? Well,
what we want is in the three cases, we
want the angle between the first basis and
the second basis to be small, so that
cosine square theta is close to one. And
in the last case, where X=1 and Y=1, we
want the angle to be close to Pi by two,
so, that cosine square theta is close to
zero. So, there. So, we got different
results in this, in this fourth case. So,
let's just list out the cases, and see how
well we do. Well, we can look at X equal
to, or let's, let's look at the case where
X, Y equal to, and what's the probability
of match? Okay. So, if X=0, Y=0, well,
then, yeah, we are measuring in the black
pieces on both sides, and the angle
between them is exactly Pi by eight. So,
the probability of match is cosine square
Pi by eight. If X=0, Y=1, then we are
measuring black here and blue there. But
the angle between these is also Pi by
eight. So, the probability of matches
cosine squared Pi by eight. What if X=1,
Y=0? Then, we have blue here and black
here. But look, the angle is still Pi by
eight. So, the probability of match is
cosine squared Pi by eight. Okay. That's a
very nice choice of, choice of basis,
because we don't have to, we, we just have
this one number to calculate. What about
the fourth case? Well now, we have X=1 and
Y=1. In this case, we are measuring in the
blue basis. So, the angle between them is
Pi by four plus Pi by eight. So, it's
three Pi by eight. So, the chance of match
is cosine squared three Pi by eight, which
is the same as, sine squared Pi over two
minus three Pi by eight. This is the same
as, sine squared, with Pi over two minus
three Pi by eight is also, Pi over eight.
But in this case, we succeed not when they
match, but when they don't match, So, so
in this case, whats the probability of not
match? And the probability of not match in
this case is, one minus sine squared Pi by
eight which is, cosine squared Pi by
eight. So, this was a very clever choice
of basis because in each of these four
cases, we succeed with probability exactly
cosine squared Pi by eight. So, the total
success probability of this particular
strategy, this, this way of carrying out
the experiment, so, probability of
success, is exactly cosine squared Pi by
eight, which if you work out, is
approximately .85. And this is the content
of Bell's theorem. Right? Okay. Now, let
me just say one more thing which might
make things a little more clearer, which
is, what did we do here? Well, what we did
here was, it says though we, we put four
numbers on the, on the number line. So, we
put the four X, wee put the, so, this was,
this was X0. If, if X=0. And so, let me
write this in blue. If this was X=1. And
then we put a point down for Y=0, and this
was, and then we put down a point for Y=1.
And we let the distances between each of
these, we let each of these distances be
Pi over eight. So, think of these as the
angles that we had. So, these were Pi over
eight. And now, what we're saying is there
are four cases. Either X=Y=0, the distance
is Pi over eight, or X is zero, Y=1 Pi
over eight, or Y is zero, X is one Pi over
eight, or X is one, Y is one, three Pi
over eight. Okay. So, this is, this is all
we did in the, in setting up those angles.
Okay. For some of you, this might help,
make things clearer. If it doesn't, just
ignore this. So, what did we actually see
in all these? Well, to recap, John Bell
devised this remarkable experime nt which
has one of two outcomes. Either, you do
this experiment, you set it up exactly the
way, you know, quantum mechanics predicts.
If the success probability ends up being
less than or equal to three quarters, then
you have to conclude that nature is
inconsistent with quantum mechanics. But
it's consistent with some local hidden
variable theory, or you do the experiment,
and the success probability is strictly
greater than three quarters, to within
your area parameters, and then you have to
conclude that nature is inconsistent with
any local hidden variable theory.
Although, it might be consistent with
quantum mechanics. The Bell experiment has
been preformed numerous times. The results
have always been consistent with quantum
mechanics. There are still a few skeptics
who say, well, you know, there are error
bias, and if you were to assume, you know,
because the detectors are not perfectly
efficient, you don't have sources of
single photons that are perfectly
reliable, and if you assume that all these
factors conspire against you in the worst
possible way, then it's, it's still
possible that quantum mechanics might be
incorrect. That, that it might still be
consistent with some sort of other theory.
But although, dealing with these, with,
with these kinds of loopholes is an
interesting questions, at this point, one
would have to say that, for all practical
purposes, you know, these loopholes are so
contrive, that we may as well believe that
local hidden variable theories have not
been ruled out by experiment. Here's one
last word, which is Einstein spent the
last twenty years of his life, trying to
show that there is a local hidden variable
theory that is consistent with everything
we know in quantum mechanics, which is,
which is consistent with everything we
know about the nature. It wasn't until
about ten years after his passing that,
John Bell came up with this, with this
experiment, but the remarkable thing to
realize is that we're already in the
fourth lecture, in the second week of
your, for many of yo u, first encounter
with quantum mechanics. You can already
understand something which is simple
enough and yet it is something that eluded
Einstein himself and which if he had
understood would have saved him twenty
years of work. Isn't that a remarkable
thing to think about?