Say Alice and Bob are on opposite sides of the universe.
Entanglement means that somehow, Alice measuring her particle affects Bob’s one even though
it’s a world away immediately.
If that’s the case, then the question of this video is “why can’t they use quantum
mechanics to communicate instantly”?
This video is very much a sequel to a video I made about entanglement and the EPR paradox
so it will help a lot to watch that first.
Actually, these videos are part of a series on quantum mechanics so start here if you’re
new.
Back to the question.
Let’s say Bob wants the answer to a yes or no question that he’s asked Alice, but
she’s waited till she’s very far away to think of an answer.
However luckily, they have an entangled pair of particles between them.
Here’s a possibility.
Alice tells Bob in advance that if he measures his particle to be up, the answer is yes,
down is no.
Now all Alice has to do is nonlocally influence his particle to be up, say and then they’ve
communicated faster than light!
But obviously that won’t work, no matter what she does, her own outcome is random,
and since Bob just gets the opposite, his outcome just comes out completely random as
well.
So that scheme doesn’t work- but a lot of you suggested a much more devious plan.
I’ve been saying for ages that you can tell the difference between a state in a superposition
of two possibilities, and one that’s merely one of the other of these but you don’t
know which.
Why not use that?
If Alice measures, the whole state collapses and is not in a superposition anymore.
Then surely Bob can do some interference experiment and if he sees no interference, he knows that
Alice has done her measurement, otherwise she hasn’t.
Alice can communicate what she wants by choosing whether to measure her particle or not.
Then they could use this scheme, and communicate instantly across the universe!!
Ok, that sounds great but obviously Quantum mechanics has a way around that too.
I’ll explain it, but this is the punch line.
With this entangled state, you can tell they’re in a superposition by doing some measurement
that shows interference.
However, that measurement is a measurement where you have to interact both of the particles
in some way to do the measurement, and so you need both particles in the same place.
If you instead only do the measurement on one of the particles, or both particles, but
not together, there’s no way for you to tell the difference between the state where
they’re in superposition or when they’re not.
To see this, let’s look at what happens when Bob does his measurement in the case
where Alice did hers already, or choose not to.
These are the states in either case.
In the case where Alice has measured, his particle really is up or down now, but he
doesn’t know which.
But then say Bob does a measurement of the left and rightness of a particle.
We know that if we did this to a particle that’s up, it will go either left or right
randomly.
Same thing for a down particle.
So even though we don’t know which it is, up or down, we know it will randomly go left
or right, and this is the no interference result that we talked about in some previous
videos.
This is kind of what we expected.
But what happens in the second case where the state is still in a superposition?
This is a bit tricky.
What you need to do is write the state, in particular Bob’s particles state, in the
left/right basis.
Try it and then see if you can figure out what happens.
OK, so here’s the maths.
All I’ve done is subbed in these expressions.
Then, I’m grouping up the left and right options for Bob’s particle.
Then you can see that both options have the same weight, so they’re both equally likely.
What this means is, if Bob measures his particle for left rightness when Alice hasn’t measured,
the particle is equally likely to go left or right- which is the same as if she had
measured, so you can’t tell the difference.
But you might protest that I’ve only showed you that this plan for faster than light communication
is foiled for this particular entangled state, but what about other ones.
There are way way way more possible entangled states that just this.
I just use this for an example most of the time because this state has the maximum possible
amount of entanglement, but in fact most 2 particle states are entangled.
For example, this state is entangled.
So what would we need to show that using any entangled state is useless for faster than
light communication?
You need to prove that no matter what Alice does on her side, that doesn’t affect Bob’s
measurement outcomes.
For example, in either case when Alice does or doesn’t choose to do a measurement, Bob
get’s up or down with the same probability either way.
This can be proved generally, but it’s a bit tough.
Instead I’m going to get you to do it for an example that’s quite a bit more challenging
that the one I did.
Prove it for this state, when Alice measures up and down-ness, but Bob measures left-rightness.
I’m going to write a detailed solution in the description, but ask me if you get confused.
So yes, entanglement is useless for faster than light communication, but can we try use
it for something else?
Say Alice and Bob have lots of entangled particles between them.
If they measure them all, then the outcomes are random, but they know what the other persons
outcome is.
So they share a string of random bits.
Your second homework is to look up and explain the usefulness of something called a ‘one-time
pad’.
Also, comment on whether or not an encryption based on a one time pad coming from this set
up is secure or not if they know Eve might be trying to intercept the particles.
This is the topic for the next video.