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The concept of entanglement was introduced
by Einstein, Podolsky, and Rosen in the

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context of a beautiful thought experiment,
where they try to show that quantum

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mechanics as it's standardly formulated,
was, is incomplete. That there's a, you,

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I'm sure you know that Einstein did not
like the, this aspect of quantum mechanics

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that, that outcomes are probabilistic,
outcomes of measurements are

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probabilistic. I'm sure you've heard the,
his quote, that God does not play dice

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with the universe. So, this thought
experiment which is called the EPR Paradox

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was introduced to show that, to show that
there is something wrong with quantum

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mechanics the way it is formulated and
that it is incomplete, that its part of a

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larger truth a, a, a more complete theory.
For our purposes in understanding

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entanglement better, it will be useful to
go through this reasoning of, of the EPR

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paradox. Okay, so let's, let's start with
where we were. So, we, we already talked

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about the Bell State which is the
superposition of 00 and eleven. And then

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we talked about measuring the Bell State
which seemed a bit paradoxical because if

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you measure the first qubit, we'd see zero
and one with equal probability, but if we

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see a zero, then the new state is, of the
system is 00, and if you see a one then

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the state is eleven which means that now
if the second particle is measured, no

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matter how far it is from the first
particle, the outcome is exactly the same,

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the outcome of the measurement is exactly
the same as the outcome of the first

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measurement. So, this should be a little
disturbing because these particles can,

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can be arbitrarily far apart and the
measurements can be arbitrarily closely

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coordinated to be almost instantaneous or
to be space-like separated so that even

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light did not have time to go from one to
the other. So, how could these two

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particles, how could these two qubits
coordinate the, the outcomes of these

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measurements so perfectly despite being so
far separated? Well, as we said in the

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last video, you could, you could imagine
that the pa rticles do this by

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coordinating their actions before they
were separated, by actually flipping a

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coin and deciding that the outcome is
going to be zero or one based on whether

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the coin came up heads or tails and using
that same coin flip, both of them, so they

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are perfectly coordinated. But now, lets
look at a different property of this Bell

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State which is if what happens if instead
of measuring in the standard basis, in the

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zero, in the, in the bit basis, we measure
in the sign basis. And this is where

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something very strange happens. So, it
turns out that the Bell State, which you

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can write as an equal super position of 00
and eleven, you can equally express it as

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an equal superposition of +,, +, and -,,
-. So, we'll see this in a moment. But,

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before we do that, let's see what, what
this implies, you know, for measurements

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of these two particles. So now, what this
means is, if we were to measure in the,

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sign basis, well, the first particle is in
the plus state with probability half and

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minus state with probability half. But, if
it happens to be measured in the plus

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state, then the other qubit will be in the
plus state with probability one. And if

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it's measured in the minus state, then the
other qubit will be in the minus state

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with probability one. So now, let's, let's
go back and see how could these two

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particles coordinate their actions so
perfectly both in the bit and in the sign

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basis simultaneous, together. Well, one
way they'd have to, they could do it, is

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of course, faster than light
communication. Because these two particles

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are so far apart and we are measuring them
at the same time essentially so that light

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did not have time to travel from one to
the other in the interval between the,

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between the measurement of the first
particle and the measurement of the second

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particle. Okay, you can see why Einstein
might object to that. What's the other

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possibility? Well, the other possibility
is that when the two particles put

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together, they could have flipped two
coins. One for the bit basis and one for

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the sign basis. And th en they agreed
that, let's say, if the coin, that maybe

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they agreed that for the bit basis, they
would answer zero and for the sign basis,

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they would answer minus. What's wrong with
that? Well, what's wrong with that is

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uncertainty principle. You see, the
particles are not allowed to know it's,

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if, if the bit value is perfectly
determined, then the sign value is

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maximally uncertain. This is what we
showed in the last lecture in the

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uncertainty principle. And so, we are in a
bind, neither of these two possibilities

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seems reasonable, relativity rules out the
first and the uncertainty principle rules

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out the second possibility. Let's look at
it another way, one way we could carry out

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this, this experiment is we could measure
the first qubit in the bit basis and the

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second qubit in the sign basis and let's
see we do these, these two measurements so

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close together that light didn't have time
to travel from one to the other. So now,

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when we measure the second cubit in the
sign basis, we know from this, this fact,

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that if the sign value turned out to be
minus, then the sign of the first qubit is

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also minus. On the other hand, when we
measured the first cubit and it turned out

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to be in the bit basis as zero, well, we
know that the first cubit must be now both

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in the bit basis, it's in the, in the
state zero. But also in the sign basis,

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it's, it's in the state minus. Surely,
this contradicts the uncertainty

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principle. So, these were the difficulties
that EPR tried to highlight with their

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gedanken experiment or their thought
experiment which is now been named EPR

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paradox. And what they believed was that,
in fact, the way out of all these was that

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the two particles gave to them all the
information necessary to locally decide

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the outcome of any future measurements. In
other words, possibly, you know, they,

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they believe that this uncertainty
principle was actually incorrect. And

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Einstein spent the rest of his life really
looking for this, a hidden variable

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theory, a local hidden variable theory for
quantum mechanics. Now, what does standard

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quantum mechanics say about all this?
Well, the big quantum mechanics, what

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quantum mechanics says is that, as soon as
the first qubit is measured, let's say in

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the bit basis, the entanglement between
the two cubits is destroyed. So that,

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because the new state is either 00 or
eleven. In each of these two states, you

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can say exactly what the state of the
first qubit is and the state of the second

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qubit is. So, if the state is 00, the
state of the first qubit is zero and the

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state of the second qubit is zero. So,
these are completely an entangled states.

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Now, if you measure the second qubit in
the sign basis, it no longer reveals any

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information about the first qubit. So, if
you were to, if you were to accept quantum

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mechanics, the rules of quantum mechanics,
there's no problem with all this. It's

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just that the rules of quantum mechanics
seem so strange, you know, this notion of

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entanglement seems so strange. The other
thing you can say about this, the rules of

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quantum mechanics is, well, the other
thing you could, you could complain about,

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is that it appears as though the two
particles are coordinating, they're

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communicating information between them.
Even if these two particles are in an

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entangled state, how could they
instantaneously come up with the same

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answer randomly? How could they randomly
choose zero versus one and + versus -? So,

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the answer to this is that, in fact, what
the two particles are achieving is some

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sort of correlation between the two sides.
But you cannot use this correlated

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outcomes to actually transmit any
information from the one particle to the

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other. In other words, suppose that you
were sitting at the site of the first

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particle and you had a bit of information,
let's say the bit was happened to be one

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and you wanted to communicate it to
somebody in a, in a different country. And

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you happen to hold one-half of this Bell
State and your friend in this other

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country held the other half of the Bell
State. What you could try to do is

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communicate this, this bit that you have
to your friend in thi s other country. We

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are this measurement of the Bell State.
And so, if you measure your qubit in the

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bit bases and if it turns out, turns out
to be one, then your friend, his qubit if

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he measures it, would also say one. But of
course, this happens only the probability

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half. With probability half and you do
your measurement, you get outcome zero,

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which is what you did not want. And then
your friend will hold a zero. So, he gets

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a, a bit which is 50, 50, zero, one. And
he gets no information at all about the

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bit you are trying to transmit to him. And
so, you, what you can show is that, in

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fact, through this means, through the
means of entanglement, it's impossible to

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transmit any information from one party to
another. All you can do is you can achieve

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this kind of coordination where you flip
the same random bit. And this doesn't

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violate the theory of relativity or speed
of light considerations. Now, Einstein

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believe to the end of his life, which was
about twenty years after he published this

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EPR paradox that there was a local hidden
variable theory that quantum mechanics was

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incomplete and he spent twenty years
searching for this local hidden variable

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theory. As it turned out in the 1960s,
John Bell, the physicist who, whom we name

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Bell States after, discover that there is
a, there is an actual experiment you can

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perform, which would demonstrate either
that quantum mechanics is incorrect or

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that there's no hidden variable theories
consistent with the way nature behaves.

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Soon afterward, these experiments were
carried out and they're consistent with

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quantum mechanics and inconsistent with
hidden variable theories. So, in the next

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lecture, I'll describe to you what this
experiment is and the principles behind

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it. It's actually quite remarkable that,
already in the second week of this, of a

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class with no background in quantum
mechanics, you can describe in full

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detail, this experiment which would have
saved Einstein twenty years of his life

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thinking about an alternative to quantum
mechanics.