BARTON ZWIEBACH: Let's talk
now about entanglement. So we talk about
entanglement when we have two non-interacting particles. You don't need a strong
interaction between particles to produce entanglement,
the particles can be totally non-interacting. Suppose particle 1 can be
in any of these states-- u 1, u 2. Let's assume just u 1 and u 2. And particle 2 can be
in states v 1 and v 2. And you have these two
particles flying around, these are possible
states of particle 1 and possible states
of particle 2. Now you want to describe
the full system, the quantum state of the two particles. States of the two particles. Two particles. Well, it seems reasonable
that to describe the state of the two particles
that are not interacting, I should tell you what
particle 1 is doing and what particle 2 is doing. OK, so particle 1
could be doing this. Could be u 1. And particle 2
could be doing v 1. And in a sense, by
telling you that, we've said what
everything is doing. Particle 1 is doing u 1,
particle 2 is doing u 2. And mathematically, we like
to make this look like a state and we want to write
it in a coherent way. And we sort of multiply
these two things, but we must say
sort of multiply, because this strange
multiplication, this, you know, we think of
them as vectors or states, so how do you multiply states? So you put something
called the tensor product, a little
multiplication like this. So you could say, don't worry,
it's kind of like a product, and it's the way we do it. We don't move things across,
the first state here, the second state here, and
that's a possible state. Now, I could have
a different state. Because particle 1, in
fact, could be doing something a little different. Could be doing alpha 1
u 1 plus alpha 2 u 2, and maybe particle 2 is doing
beta 1 v 1 plus beta 2 v 2. And this would be all right. I'm telling you what particle
1 is doing and I'm telling you what particle 2 is
doing and the rules of tensor multiplication or
this kind of multiplication to combine those states
are just like a product, except that as I said, you
never move the states across. So you just distribute, so
you have alpha 2, beta 1, the number goes out, u 1 v 1-- that's the first factor-- plus alpha 1 beta 2 u 1 v 2
plus alpha 2 beta 1 u 2 v 1 plus alpha 2 beta 2 u 2 v 2. I think I got it right. Let me know. I just multiplied and
got the numbers out. The numbers can be move
out across this product. OK, so that's a state and that's
a superposition of states, so actually, I could try to
write a different state now. You see, we're just
experimenting, but here is another state. u 1 v 1 plus u 2 v 2. Now this is a state that
actually seems different. Quite different. Because I don't seem
to be able to say that what particle
1 is doing and what particle 2 is doing separately. You see, I can say when
particle 1 is doing u 1, particle 2 is doing v 1. And if when particle 1 is
doing u 2, this is v 2. But can I write
this as some state of the first particle times some
state of the second particle? Well, let's see. Maybe I can and can
write it in this form. This is the most general
state that you can say, particle 1 is doing this,
and particle 2 is doing that. So can they do that? Well, I can compare these
two terms with those and they conclude that
alpha 1 beta 1 must be 1. Alpha 2 beta 2 must be also 1. But no cross products exist,
so alpha 1 beta 2 must be 0 and alpha 2 beta 1 must be 0. And that's a problem
because either alpha 1 is 0, which is
inconsistent, or beta 2 is 0, which is
inconsistent with that, so now this state
is un-factorizable. It's a funny state in which you
cannot say that this quantum state can be described by
telling what the first particle is doing and what the
second particle is doing. What the first particle is
doing depends on the second and what the second is
doing depends on the first. This is an entangled state. And then we can build entangled
states and our very strange states. So with two
particles with spins, for example, we can build
an entangled states of 2 spin 1/2 particles. And this state could
look like this-- the first particle is up along
z and the second particle is down along z, plus
a particle that is down along z for
the first particle, but the second is up along z. And these are 2
spin 1/2 particles and in the usual notation,
these experiments in quantum mechanics and
black hole physics, people speak of Alice and Bob. Alice has one particle,
Bob has the other particle. Maybe Alice is in the
moon and has her electron and Bob is on earth
and has his electron, and the two electrons, one
on the earth and in the moon, are in this state. So then we say that Alice and
Bob share an entangled pair. And all kinds of
strange things happen. People can do those things
in the lab-- not quite one in the earth and
one in the moon, but one photon at one
place and another photon entangled with it at
100 kilometers away, that's pretty doable. And they are in this funny state
in which their properties are currently that in
surprising ways. So what happens here? Suppose Alice
goes-- or let's say Bob goes along and
measures his spin and he finds his spin down. So-- oh, you look here,
oh, here is down for Bob. So at this moment, the whole
state collapses into this. Because up with Bob
didn't get realized. So once Bob measures
and he finds down, the whole state goes into this. So if Alice-- on the moon
or in another galaxy-- at that instant
looks at her spin, she will find it's
up before light has had time to get there. Instantaneously. It will go into this state. People were sure somehow this
violates special relativity. It doesn't. You somehow when you think
about this carefully, you can't quite
send information, but the collapse
is instantaneous in quantum mechanics. Somehow, Bob and Alice cannot
communicate information by sharing this entangled pair,
but it's an interesting thing why it cannot happen. Einstein again objected to this. And he said, this
is a fake thing. You guys are going to share-- and now, of course, they have
to share many entangled pairs to do experiments, so maybe
1,000 entangled pairs. And Einstein would say, no,
that's not what's happening. What's happening is that some of
your entangled pairs are this. That is, Bob is down, Alice
is up, some of them are this-- and there's no such thing
as this entanglement and indeed, if you
measure and you find down, she will find up, and if
you measure and you find up, she will find down, and there's
nothing too mysterious here. But then came John Bell in
1964 and discovered his Bell inequalities that demonstrated
that if Alice and Bob can measure in three
different directions, they will find correlations
that are impossible to explain with classical physics. It took a lot of originally
for Bell to discover this, that you have to measure
in three directions, and therefore, the
kind of correlations that appear in entangled states
are very subtle and pretty difficult to disentangle. So that's why entanglement
is a very peculiar subject. People think about it a lot
because it's very mysterious. It somehow violates
classical notions, but in a very subtle way.
