From the decoherent point of view, the no-communication theorem is fairly simple (if you are comfortable with tensor products*). Suppose that Alice and Bob are studying the two quantum systems A and B, whose state spaces are represented by Hilbert spaces H_A and H_B, respectively. Then the state space of the joint system is H:=H_AotimesH_B. Now suppose that Alice makes a measurement on** system A, and Bob makes a measurement on system B. These measurements are represented physically by unitary transformations U_A:H_ArightarrowH_A and U_B:H_BrightarrowH_B. The effect of the measurements on the joint system are therefore represented by the unitary transformations V_A=U_AotimesI_B and V_B=I_AotimesU_B, where I_A and I_B are the identity transformations on H_A and H_B, respectively. The key to the no-communication theorem is the observation that the transformations V_A and V_B commute with each other. (Either way you take the product you get U_AotimesU_B.) It implies that if we do our calculations assuming that Alice did her measurement first, then we will get the same answers as if we do our calculations assuming that Bob did his measurement first. So let’s do our calculations assuming that Bob measured first, as it will be easier to analyze that way.
After Bob makes his measurement, the amplitude of the universe is split up into two blobs, one corresponding to Bob recording Possible Outcome 1 and another correspondint to Bob recording Possible Outcome 2. The size of these blobs, as measured by square-integrating, is independent of anything that Alice does (since according to this formulation of the problem, Alice hasn’t done anything yet). Now when Alice makes her measurement, the size of the blobs is preserved because of unitarity. Moreover (and this is the crucial point) the blob corresponding to Outcome 1 gets mapped to another blob corresponding to Outcome 1, and the blob corresponding to Outcome 2 gets mapped to another blob corresponding to Outcome 2. Thus, the final size of the blobs corresponding to the different outcomes is independent of Alice’s choice, and according to the Born probabilities that means Bob’s expectations about his measurement are also independent of Alice’s choice.
The fact that outcomes are preserved under Alice’s action is worth remarking further on. Intuitively, it corresponds to the fact that recorded measurements don’t erase themselves randomly. Scientifically, it corresponds to the complicated phenomenon known as decoherence, which is much harder to describe rigorously than the no-communication theorem is. Philosophically, it corresponds to the fact about the world that the Copenhagen interpretation thinks of as an assumption, and which many-worlders think too complicated to be considered a fundamental assumption of physics.
* For those not familiar with tensor products, they are the mathematical objects Eliezer is implicitly talking about whenever he writes things like “(Human-LEFT Sensor-LEFT Atom-LEFT) + (Human-RIGHT Sensor-RIGHT Atom-RIGHT)”. A working definition is that the tensor product of an M-dimensional space with an N-dimensional space is an MN-dimensional space.
** And/or a modification to system A; the composition of any number of measurements and modifications will always be represented by a unitary transformation.
A final remark: The no-communication theorem, as I’ve sketched it above, shows that entangled but noninteracting particles cannot be used for distant communication. It says nothing about faster-than-light communication, as it does not make the connection between the ability of particles to interact and the speed of light, a connection which requires more formalism. The fact that FTL communication is impossible is a theorem of quantum field theory, the relativistic version of quantum mechanics. The basic idea is that the evolution operators corresponding to spacelike separated regions of spacetime will commute, allowing the above argument to take place with V_A and V_B replaced by more realistic operators.
By the way, has anyone else noticed that math symbols don’t always work in LessWrong markup? I originally posted code which I had compiled from LaTeX to markup at the suggested website and then double-checked the markup output at http://markdownr.com/, but when I posted here there were errors which didn’t come up on either of the previous sites. (I think I’ve fixed all the errors now though...)
This would be a lot less annoying if it were possible to preview a comment before posting it...
From the decoherent point of view, the no-communication theorem is fairly simple (if you are comfortable with tensor products*). Suppose that Alice and Bob are studying the two quantum systems A and B, whose state spaces are represented by Hilbert spaces H_A and H_B, respectively. Then the state space of the joint system is H:=H_AotimesH_B. Now suppose that Alice makes a measurement on** system A, and Bob makes a measurement on system B. These measurements are represented physically by unitary transformations U_A:H_ArightarrowH_A and U_B:H_BrightarrowH_B. The effect of the measurements on the joint system are therefore represented by the unitary transformations V_A=U_AotimesI_B and V_B=I_AotimesU_B, where I_A and I_B are the identity transformations on H_A and H_B, respectively. The key to the no-communication theorem is the observation that the transformations V_A and V_B commute with each other. (Either way you take the product you get U_AotimesU_B.) It implies that if we do our calculations assuming that Alice did her measurement first, then we will get the same answers as if we do our calculations assuming that Bob did his measurement first. So let’s do our calculations assuming that Bob measured first, as it will be easier to analyze that way.
After Bob makes his measurement, the amplitude of the universe is split up into two blobs, one corresponding to Bob recording Possible Outcome 1 and another correspondint to Bob recording Possible Outcome 2. The size of these blobs, as measured by square-integrating, is independent of anything that Alice does (since according to this formulation of the problem, Alice hasn’t done anything yet). Now when Alice makes her measurement, the size of the blobs is preserved because of unitarity. Moreover (and this is the crucial point) the blob corresponding to Outcome 1 gets mapped to another blob corresponding to Outcome 1, and the blob corresponding to Outcome 2 gets mapped to another blob corresponding to Outcome 2. Thus, the final size of the blobs corresponding to the different outcomes is independent of Alice’s choice, and according to the Born probabilities that means Bob’s expectations about his measurement are also independent of Alice’s choice.
The fact that outcomes are preserved under Alice’s action is worth remarking further on. Intuitively, it corresponds to the fact that recorded measurements don’t erase themselves randomly. Scientifically, it corresponds to the complicated phenomenon known as decoherence, which is much harder to describe rigorously than the no-communication theorem is. Philosophically, it corresponds to the fact about the world that the Copenhagen interpretation thinks of as an assumption, and which many-worlders think too complicated to be considered a fundamental assumption of physics.
* For those not familiar with tensor products, they are the mathematical objects Eliezer is implicitly talking about whenever he writes things like “(Human-LEFT Sensor-LEFT Atom-LEFT) + (Human-RIGHT Sensor-RIGHT Atom-RIGHT)”. A working definition is that the tensor product of an M-dimensional space with an N-dimensional space is an MN-dimensional space.
** And/or a modification to system A; the composition of any number of measurements and modifications will always be represented by a unitary transformation.
A final remark: The no-communication theorem, as I’ve sketched it above, shows that entangled but noninteracting particles cannot be used for distant communication. It says nothing about faster-than-light communication, as it does not make the connection between the ability of particles to interact and the speed of light, a connection which requires more formalism. The fact that FTL communication is impossible is a theorem of quantum field theory, the relativistic version of quantum mechanics. The basic idea is that the evolution operators corresponding to spacelike separated regions of spacetime will commute, allowing the above argument to take place with V_A and V_B replaced by more realistic operators.
By the way, has anyone else noticed that math symbols don’t always work in LessWrong markup? I originally posted code which I had compiled from LaTeX to markup at the suggested website and then double-checked the markup output at http://markdownr.com/, but when I posted here there were errors which didn’t come up on either of the previous sites. (I think I’ve fixed all the errors now though...)
This would be a lot less annoying if it were possible to preview a comment before posting it...