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Hello, everybody. Today's lecture is about
Bell's Experiment. This experiment was

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designed by the physicist John Bell in the
1960s, and this experiment has provided

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deep insights into the nature of quantum
mechanics, into the nature of

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entanglement. And, even today, it's an,
it's a topic of active research in quantum

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cryptography, quantum information, quantum
computation. Okay, so, so, let's start

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with a philosophical question. Can we
experimentally test whether we live in a

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quantum world? What do we mean by this?
Well, all the quantum weirdness, the fact

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that measurement outcomes are
probabilistic, that just looking at the

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system changes it's state. The fact that
particles do not have trajectories, and

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instead, they are in this weird kind of
super position where they take all

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possible paths simultaneously at some
amplitude. Is all this an intrinsic

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property of nature or does it reflect the
fact that we only have an incomplete

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picture, that, that quantum mechanics is
an incomplete theory? That is, if you knew

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more about the mechanism behind the
scenes, if you knew more about the gears

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and, and pulleys that are acting behind
the scenes, would it actually look much

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more natural? This is what Einstein
believed. Okay, so the kind of mechanism

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that Einstein was looking for, that acts
behind the scenes, was a local and

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deterministic mechanism. By local, what
that means is, that there is no action at

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a distance. Of course, local also means
that there's no faster than light

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communication an the theory of relativity
holds. And deterministic, meaning that

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there are no, no such probabilistic
outcomes to measurements that the system

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actually has definite values for physical
quantities like, momentum or speed, or

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spin, etc., or, or, or bit or sine, etc.
I'm sure you've heard the quote,

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Einsteins's quote, God does not play dice
with the universe. Okay, so this is what,

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what we are looking for and in the
previous videos we talked about the EPR

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experiment, the Einstein, Pedalski, Rosen
thought experiment, which was designed to

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precisely think about these issues. Now in
the 1960's, in 1964, John Bell designed a

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remarkable experiment. This was an honest
to goodness experiment, not just a thought

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experiment. And, it was an experiment
which had one of two outcomes. So, if you

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do this experiment, and the outcome is
outcome one, we'll see what all these

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outcomes mean later. But if, if, if the
first outcome happened , then you could

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conclude that nature is inconsistent with
quantum mechanics. That quantum mechanics

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is false. But that nature can be explained
by some kind of local hidden variable

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theory. It's consistent with some local
hidden variable theory. On the other hand,

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if the second outcome holds, then this
would imply that nature is consistent with

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quantum mechanics, but it's inconsistent
with any local hidden variable theory.

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Meaning, that if the second outcome were
true, then the kind of theory that

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Einstein had spent the last twenty years
of his life searching for, could not

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possibly hold. It would have been
experimentally proved that, in fact all

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such theories are ruled out by the way
nature behaves, can be observed to behave.

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Now, the Bell experiment relies on
sophisticated properties of entangled

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cubits. Actually of, of the, of the Bell
state. And the Bell experiment has been

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performed numerous times. And the results
have always have been consistent with

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quantum mechanics, and inconsistent with
local hidden variable theories. So, always

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it's outcome two within, within
experimental error that's, that's held.

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Okay, let's look at Bell's experimental
setup So, Bell assumed that the apparatus

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for this experiment is divided into two
parts which are far enough apart, that

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during the course of the experiment, light
does not have enough time to travel from

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one to the other. So, the two pieces are
completely isolated from each other. Of

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course, this can be achieved by making
the, the duration of the experiment so

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short, that you could actually have these
two parts of the apparatus at two ends of

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your, your lab. Okay, so now, for all
purposes, let's think of these two parts

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of the apparatus as two boxes. The first
box gets us input. One of two, you know

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bit values 01, which we'll call X. And the
second box, gets us input a bit, which

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we'll call Y. What the boxes are supposed
to do is, they are supposed to output also

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a bit, zero or one. And what they are
trying to do is, if both the input bits, X

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and Y are one, then the output bits, A and
B are required to be different from each

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other. So, they could be either 01 or ten.
In all the other three cases, we want the

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outputs of the two boxes to be the same.
Either 00 or eleven. Now, what Bell was

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able to show is that, if the boxes, if the
physics of the boxes is described by local

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hidden variable, local hidden variable
theory, then they cannot succeed with, in

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this task with probability exceeding three
quarters or .75. So, no matter what the

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physics of these boxes is, no matter what
the physical theory is, as long as it's

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local, it's a local hidden variable
theory, the chance of success is at most

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three quarters. On the other hand, he was
able to show that, if the two boxes share

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a bell state, meaning that there's one
cubit in the first box, and a cubit in the

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second box, so, that these two cubits are
entangled, and they form this bell state

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00 plus eleven. Then this, there are
certain measurements that you can make on

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these two cubits, that the two cubits can
make, in such a way that they succeed in

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this task with probability as high as .85.
And so, if you perform such an experiment,

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if you actually realize this experiment
and if within your error tolerances, you,

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you showed that your success probability
in this task is strictly greater than .75.

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Then you'd have shown that nature is
inconsistent with any local hidden

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variable theory, and it's consistent with
quantum mechanics. Actually, to show that

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it's consistent with quantum mechanics,
you'd have to show that, that you're

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success probability to within your error
of tolerance is, is really .85. Okay. So,

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let's, let's now see, why a local hidden
variabl e theory does not allow you to

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succeed in this task with probability
greater than three quarters. So, the fact

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that its a local hidden variable theory,
and the fact that these two boxes cannot

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communicate with each other, what this
implies is that output of the first box

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can depend on X, but it cannot depend on
Y, the input to the second box. And

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similarly, the output of the second box
cannot depend upon X, the input of the

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first box. Also, the fact that we want to
show that the success probability cannot

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be greater than three quarters. For
contradiction, let's assume that the

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success probability is greater than three
quarters. Well, in this case, the boxes

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cannot afford to be incorrect on any of
the four possible inputs 00, 01, etc, etc.

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Okay. So, now suppose, so, what this mean
is that, if the input was X=0 and Y=0,

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then the two output bits must be the same.
A must be equal to B. So, let's assume

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without, without loss of generality that,
A = B = zero in this case. Now, by

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locality, if we were to switch only one of
the inputs, let's say Y and switch Y, make

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Y=1 while leaving X=0. So, now the first
box will still output A=0. But since the

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two boxes must output the same value on
this particular input, X=0, Y=1. So, in

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fact, the second box must also output B=0.
We can use the same reasoning if, instead

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we leave the input Y=0 and now switch X=1.
So, we, we now reason that, well, of

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course, B still must be zero, but now,
since A must be equal to B, in fact, in

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this case, also A=B=0. Okay. Now, let's
argue that this further implies that when

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we gave the two boxes, the inputs X=1,
Y=1, they still must output A=0 and B=0.

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But this is a contradiction, because in
this case, they were supposed to output

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different bits. So, they were supposed to
output either A=0 an B=1 or A=1 and B=0

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So, that's a contradiction. So, let's see
how this last step works. This is a little

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subtle. Okay. So, you see, in the case
X=0, Y=1, the outputs were, the first box

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output zero, and the second box output
zero. Well, so, let's pay attention to the

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second box in this case. Well, from the
second box's point of view, it was getting

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us input Y=1. It doesn't know whether the
first box got input X=0 or X=1. So, it

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must get the same output regardless of
which input it, the first box got. But in

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the case that the first box got input
zero, the second box output B=0. So, it

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must still output B=0 in the case that
X=Y=1. Symmetrically, we can also argue

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that from the first box's point of view,
when it was given input one, it must

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output A=0, because it doesn't know
whether it's in the case X=1 and Y=0 or

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whether it's in the case X=1, Y=1. So, we
have argued that in the case X=Y=1, both

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boxes must still output zero and so, they
are both outputting the same bit, which is

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a contradiction. Okay. So, so we've proved
our first assertion here that if the boxes

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are described by local hidden variables,
they succeed with probability at most

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three quarters. The second assertion that
if the boxes actually follow quantum

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mechanics, if they are allowed to share a
bell state, then they can succeed with a

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probability as high as .85. This is
actually somewhat more involved, and to do

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this we need to understand some properties
of bell states. Which we'll do in the next