# Does entanglement not immediately contradict the theory of special relativity?

Does entanglement not immediately contradict the theory of special relativity? Why are people still so convinced nothing can travel faster than light when we are perfectly aware of something that does?

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Possible duplicate: physics.stackexchange.com/q/3158/2451 and links therein. –  Qmechanic Apr 14 '13 at 23:40
–  Ben Crowell Apr 15 '13 at 0:25

To answer this kind of question properly, it's important to clarify the foundational issues of why SR forbids superluminal speeds and what kind of superluminal speeds it forbids. There are several independent arguments of this kind that tell us several different things.

1. Superluminal transmission of information would violate causality, since it would allow a causal relationship between events that were spacelike in relation to one another, and the time-ordering of such events is different according to different observers. Since we never observe causality to be violated, we suspect that superluminal transmission of information is impossible. This leads us to interpret the metric in relativity as being fundamentally a statement of possible cause and effect relationships between events.

2. We observe the invariant mass defined by $m^2=E^2-p^2$ to be a fixed property of all objects. Therefore we suspect that it is not possible for an object to change from having $|E|>|p|$ to having $|E|<|p|$.

3. Composing a series of Lorentz boosts produces a velocity that approaches $c$ only as a limit. Therefore no continuous process of acceleration can bring an observer from $v<c$ to $v>c$. Since it's possible to build an observer out of material objects, it seems that it's impossible to get a material object past $c$ by a continuous process of acceleration.

4. If we could boost a material object past the speed of light, even by some discontinuous process, then we could do so for an observer. However, there is a no-go theorem, Gorini 1971, proving that this is impossible in 3+1 dimensions.

Entanglement doesn't violate any of these arguments. It doesn't violate #1, since it doesn't transmit information. It doesn't violate #2, #3, or #4, since it doesn't involve boosting any object past the speed of light.

V. Gorini, "Linear Kinematical Groups," Commun Math Phys 21 (1971) 150, open access at http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1103857292

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I have never understood why entanglement doesn't transmit information faster than the speed of light. I alway imagine two entangled, polarized photons, one going right and the other going left. Then you'd measure one to force the other to the same state. It seems like you could transmit information faster than the speed of light this way (or some variation on it). Why can't you? Do you have a layman's explanation or a link to one? –  Brandon Enright Apr 14 '13 at 23:56
@BrandonEnright Because the outcome of a quantum measurement is random and uncontrollable, and there is no way for the receiver to know whether the state has changed or not without doing a measurement themselves. –  Michael Brown Apr 15 '13 at 0:03
@BrandonEnright: That's more of a separate question. At the simplest level, I like this analogy: physics.stackexchange.com/a/3162/4552 –  Ben Crowell Apr 15 '13 at 0:15
@BenCrowell, be careful, because that analogy with gloves is not correct: it is exactly the kind of local hidden variable model that is provably inconsistent with the predictions of quantum mechanics via Bell's theorem. –  Mark Mitchison Apr 15 '13 at 0:27
@BenCrowell Yeah I'm sure you understand the problem, and I don't want to get drawn into this but I'll just say that the top answers and comments on the page you linked are only really sensible from an anti-realist point of view. Any readers that believe in a reality existing independently of observers (what is an observer, exactly?) should treat the comments and answers given on the linked page with caution. –  Mark Mitchison Apr 15 '13 at 0:41

Spooky action at a distance is faster than light Chinese boffins put the clock on information transfer between entangled particles By Richard Chirgwin Posted in Science, 8th April 2013 05:09 GMT

As Einstein put it, it's impossible for anything – even information – to move faster than the speed of light. Yet the lower bound of that impossibility, the minimum speed at which entanglement can't possibly be transmitting information between two particles, appears to be around four orders of magnitude higher than $c$, the speed of light in a vacuum.

http://www.theregister.co.uk/2013/04/08/chinese_entanglement_transfer_experiment/

http://arxiv.org/abs/1303.0614

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Here's an example I like of why entanglement doesn't let you violate relativity. Say you have two spaceships moving in opposite directions along a line, with constant velocity. At $t = 0$, they synchronize clocks and entangle two particles. They also decide, at some predetermined time $T$, to measure the spins of the particles (actually, ship 1 will measure at time $T$, and ship 2 will measure at $T + \epsilon$). They will interpret these measurements as ship 1 picking out a definite value for the spin of their particle (and thus of its entangled partner) and ship 2 measuring this value by comparing their own measured spin with the initial entangled state. If this worked, it would seem the spin information was transmitted from ship 1 to ship 2 faster than light, for sufficiently small $\epsilon$.

I like this example because you could (hypothetically) try it in real life, so there must be some concrete reason it wouldn't work. In fact, the relativity of simultaneity makes the interpretation of the measurements as information-transfer invalid. Ship 2 will make its measurement at time $T + \epsilon$, but it can't interpret the result as information sent from ship 1, because (as ship 2 can tell from its own clock plus a simple calculation) ship 1 hasn't made a measurement yet in ship 2's reference frame. The same reasoning applies in reverse; neither ship receives information about the other's measurement unless $\epsilon$ is large enough to make the simultaneity absolute.

So while there's undoubtedly more to be said about what's going on in general, this example reassures me that there's no immediate contradiction here.

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