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Why doesn't entanglement allow faster than light communication?

The reason entanglement doesn't allow for faster than light communication is due to the subtleties of quantum mechanics. I'll try to explain that by using an analogy.

Let's say Alice flies off to a planet a million light years away and she wants to tell Bob, who remains here on Earth, whether she's found life on that planet. She'll send Bob a 1 if there's life and a 0 if there isn't. The thing is, Bob doesn't want to wait a million years for the answer to reach him at the speed of light; he wants the answer instantly.

Alice knows a bit about quantum mechanics and she knows that if she entangles two particles on Earth, those two particles will remain entangled no matter how far apart those particles are. She also knows that if she measures the state of her particle, it will instantly affect the state of Bob's particle. This, she feels, is a good basis for faster than light communication.

If Alice, now a million light years away, measures her particle and sees a 1, she knows that when Bob measures his particle he'll also see a 1. So let's say they ignore the complexities of relative time and find a way to coordinate them both opening their box on Christmas Day in the year 2525. Alice will open her box at one second to noon and Bob will open his box at noon precisely. This allows Alice to perform her measurement before Bob but provides nowhere near enough time for any signals to pass at light speed.

The agreed time comes around and Alice opens her box and sees a 1. One second later, Bob opens his box and sees a 1. Bob instantly knows Alice has seen a 1 even though she's a million light years away.

There we go: faster than light communication. Except it isn't. Bob already knew that if he saw a 1 then that's what Alice would also have seen. No new information has been transmitted. Bob is none the wiser about the existence of life on the distant planet. He just knows that the 50-50 chance Alice had of seeing a 1 at the outset has been resolved. She could equally have seen a 0 — the outcome of opening Alice's box was entirely random. Sure, the outcome of opening Bob's box was correlated with Alice's, but the procedure was random at source.

Let's suppose Alice and Bob are a bit smarter. Alice needs to do more than merely measure her particle, she needs to be able to set it to 1 or 0 depending on whether she finds life on the planet a million light years away. She finds out there is a way to do that (and there is). On the faraway planet she finds a strange creature that invites her to lunch. Excitedly, she goes back to her spacecraft to tell Bob she's found life. She sets her particle to 1 and one second later Bob looks at his particle, finds a 1 and throws a party in honour of first contact.

Alas, Bob has thrown a party for nothing. When Alice forces her particle to be a 1 or 0, it breaks the correlation between the two particles. It's the subtle difference between measuring a particle and interfering with a particle. When Alice interferes with her particle, the correlation is broken and Bob's particle exists independently in a superposition of 1 and 0. He has a 50-50 chance of measuring a 1 or a 0 and it will tell him nothing.

It's back to drawing board. Alice and Bob need to find a way to communicate without breaking the correlation. That means they can only take measurements rather than changing the state of their particles.

Scrappy drawing of light polarisation. Let's get a bit more specific and instead of 'a particle', they agree to use photons. Photons can be polarised, which is how a sunglasses work. Only the photons polarised in a certain way will get through the sunglasses. Let's assume that if the light's polarised vertically it gets through and if it's polarised horizontally it does not get through. In the middle — at 45 degrees — there's a 50% chance the light will get through and a 50% chance it won't.

Alice clears off to the distant planet and the idea is that if she finds life she'll measure her particle's polarisation in either the horizontal or vertical plane, signifying a 1. If she does not find life she'll measure her particle's polarisation at 45 degrees, signifying a 0. Nothing has been interfered with; she's simply measuring.

Back on Earth, Bob always measures at 45 degrees. He gains nothing by measuring a single particle. If Alice sets a 1, Bob will measure a 1, but if Alice sets a 0, Bob has a 50-50 chance of measuring either 1 or 0. With one measurement he won't know it he's got a definite result or just got lucky. And vice versa: if Alice sets a 0 there's still a 50-50 chance in Bob's measurement.

Aha, they think, what they could do is clone Bob's particle 100 times and look at a probability distribution. If they could make more than one measurement they could eradicate Bob's original 50-50 chance. When he counted things up he would see they are leaning towards a particular result.

Have they cracked it? In theory yes, but how are they going to clone Bob's photon? The whole principle behind their experiment is that the entangled particles are in an indeterminate state — a superposition — at the outset. Then when Alice makes a measurement, that indeterminate state collapses and Alice's particle takes a precise state, and in doing so it also sets Bob's photon to a precise state thanks to the entanglement. But how do you clone something that's in an indeterminate state? You'd need to know the exact state of it to clone it properly and in the process of finding out you'd disturb the original state[1].

Alice and Bob are very unhappy, particularly Alice who's now a million light years away, at lunch with an alien.

It's almost as if the universe conspires against us and, one way or another, prevents faster than light communication.

[1]: This called the 'no cloning' property of quantum mechanics and is due to something called 'quantum linearity' if you want to do some searches and investigate further.