The concept of entanglement was introduced
by Einstein, Podolsky, and Rosen in the
context of a beautiful thought experiment,
where they try to show that quantum
mechanics as it's standardly formulated,
was, is incomplete. That there's a, you,
I'm sure you know that Einstein did not
like the, this aspect of quantum mechanics
that, that outcomes are probabilistic,
outcomes of measurements are
probabilistic. I'm sure you've heard the,
his quote, that God does not play dice
with the universe. So, this thought
experiment which is called the EPR Paradox
was introduced to show that, to show that
there is something wrong with quantum
mechanics the way it is formulated and
that it is incomplete, that its part of a
larger truth a, a, a more complete theory.
For our purposes in understanding
entanglement better, it will be useful to
go through this reasoning of, of the EPR
paradox. Okay, so let's, let's start with
where we were. So, we, we already talked
about the Bell State which is the
superposition of 00 and eleven. And then
we talked about measuring the Bell State
which seemed a bit paradoxical because if
you measure the first qubit, we'd see zero
and one with equal probability, but if we
see a zero, then the new state is, of the
system is 00, and if you see a one then
the state is eleven which means that now
if the second particle is measured, no
matter how far it is from the first
particle, the outcome is exactly the same,
the outcome of the measurement is exactly
the same as the outcome of the first
measurement. So, this should be a little
disturbing because these particles can,
can be arbitrarily far apart and the
measurements can be arbitrarily closely
coordinated to be almost instantaneous or
to be space-like separated so that even
light did not have time to go from one to
the other. So, how could these two
particles, how could these two qubits
coordinate the, the outcomes of these
measurements so perfectly despite being so
far separated? Well, as we said in the
last video, you could, you could imagine
that the pa rticles do this by
coordinating their actions before they
were separated, by actually flipping a
coin and deciding that the outcome is
going to be zero or one based on whether
the coin came up heads or tails and using
that same coin flip, both of them, so they
are perfectly coordinated. But now, lets
look at a different property of this Bell
State which is if what happens if instead
of measuring in the standard basis, in the
zero, in the, in the bit basis, we measure
in the sign basis. And this is where
something very strange happens. So, it
turns out that the Bell State, which you
can write as an equal super position of 00
and eleven, you can equally express it as
an equal superposition of +,, +, and -,,
-. So, we'll see this in a moment. But,
before we do that, let's see what, what
this implies, you know, for measurements
of these two particles. So now, what this
means is, if we were to measure in the,
sign basis, well, the first particle is in
the plus state with probability half and
minus state with probability half. But, if
it happens to be measured in the plus
state, then the other qubit will be in the
plus state with probability one. And if
it's measured in the minus state, then the
other qubit will be in the minus state
with probability one. So now, let's, let's
go back and see how could these two
particles coordinate their actions so
perfectly both in the bit and in the sign
basis simultaneous, together. Well, one
way they'd have to, they could do it, is
of course, faster than light
communication. Because these two particles
are so far apart and we are measuring them
at the same time essentially so that light
did not have time to travel from one to
the other in the interval between the,
between the measurement of the first
particle and the measurement of the second
particle. Okay, you can see why Einstein
might object to that. What's the other
possibility? Well, the other possibility
is that when the two particles put
together, they could have flipped two
coins. One for the bit basis and one for
the sign basis. And th en they agreed
that, let's say, if the coin, that maybe
they agreed that for the bit basis, they
would answer zero and for the sign basis,
they would answer minus. What's wrong with
that? Well, what's wrong with that is
uncertainty principle. You see, the
particles are not allowed to know it's,
if, if the bit value is perfectly
determined, then the sign value is
maximally uncertain. This is what we
showed in the last lecture in the
uncertainty principle. And so, we are in a
bind, neither of these two possibilities
seems reasonable, relativity rules out the
first and the uncertainty principle rules
out the second possibility. Let's look at
it another way, one way we could carry out
this, this experiment is we could measure
the first qubit in the bit basis and the
second qubit in the sign basis and let's
see we do these, these two measurements so
close together that light didn't have time
to travel from one to the other. So now,
when we measure the second cubit in the
sign basis, we know from this, this fact,
that if the sign value turned out to be
minus, then the sign of the first qubit is
also minus. On the other hand, when we
measured the first cubit and it turned out
to be in the bit basis as zero, well, we
know that the first cubit must be now both
in the bit basis, it's in the, in the
state zero. But also in the sign basis,
it's, it's in the state minus. Surely,
this contradicts the uncertainty
principle. So, these were the difficulties
that EPR tried to highlight with their
gedanken experiment or their thought
experiment which is now been named EPR
paradox. And what they believed was that,
in fact, the way out of all these was that
the two particles gave to them all the
information necessary to locally decide
the outcome of any future measurements. In
other words, possibly, you know, they,
they believe that this uncertainty
principle was actually incorrect. And
Einstein spent the rest of his life really
looking for this, a hidden variable
theory, a local hidden variable theory for
quantum mechanics. Now, what does standard
quantum mechanics say about all this?
Well, the big quantum mechanics, what
quantum mechanics says is that, as soon as
the first qubit is measured, let's say in
the bit basis, the entanglement between
the two cubits is destroyed. So that,
because the new state is either 00 or
eleven. In each of these two states, you
can say exactly what the state of the
first qubit is and the state of the second
qubit is. So, if the state is 00, the
state of the first qubit is zero and the
state of the second qubit is zero. So,
these are completely an entangled states.
Now, if you measure the second qubit in
the sign basis, it no longer reveals any
information about the first qubit. So, if
you were to, if you were to accept quantum
mechanics, the rules of quantum mechanics,
there's no problem with all this. It's
just that the rules of quantum mechanics
seem so strange, you know, this notion of
entanglement seems so strange. The other
thing you can say about this, the rules of
quantum mechanics is, well, the other
thing you could, you could complain about,
is that it appears as though the two
particles are coordinating, they're
communicating information between them.
Even if these two particles are in an
entangled state, how could they
instantaneously come up with the same
answer randomly? How could they randomly
choose zero versus one and + versus -? So,
the answer to this is that, in fact, what
the two particles are achieving is some
sort of correlation between the two sides.
But you cannot use this correlated
outcomes to actually transmit any
information from the one particle to the
other. In other words, suppose that you
were sitting at the site of the first
particle and you had a bit of information,
let's say the bit was happened to be one
and you wanted to communicate it to
somebody in a, in a different country. And
you happen to hold one-half of this Bell
State and your friend in this other
country held the other half of the Bell
State. What you could try to do is
communicate this, this bit that you have
to your friend in thi s other country. We
are this measurement of the Bell State.
And so, if you measure your qubit in the
bit bases and if it turns out, turns out
to be one, then your friend, his qubit if
he measures it, would also say one. But of
course, this happens only the probability
half. With probability half and you do
your measurement, you get outcome zero,
which is what you did not want. And then
your friend will hold a zero. So, he gets
a, a bit which is 50, 50, zero, one. And
he gets no information at all about the
bit you are trying to transmit to him. And
so, you, what you can show is that, in
fact, through this means, through the
means of entanglement, it's impossible to
transmit any information from one party to
another. All you can do is you can achieve
this kind of coordination where you flip
the same random bit. And this doesn't
violate the theory of relativity or speed
of light considerations. Now, Einstein
believe to the end of his life, which was
about twenty years after he published this
EPR paradox that there was a local hidden
variable theory that quantum mechanics was
incomplete and he spent twenty years
searching for this local hidden variable
theory. As it turned out in the 1960s,
John Bell, the physicist who, whom we name
Bell States after, discover that there is
a, there is an actual experiment you can
perform, which would demonstrate either
that quantum mechanics is incorrect or
that there's no hidden variable theories
consistent with the way nature behaves.
Soon afterward, these experiments were
carried out and they're consistent with
quantum mechanics and inconsistent with
hidden variable theories. So, in the next
lecture, I'll describe to you what this
experiment is and the principles behind
it. It's actually quite remarkable that,
already in the second week of this, of a
class with no background in quantum
mechanics, you can describe in full
detail, this experiment which would have
saved Einstein twenty years of his life
thinking about an alternative to quantum
mechanics.