Comment author: Dacyn 03 August 2014 11:49:40PM *  0 points [-]

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 and , whose state spaces are represented by Hilbert spaces and , respectively. Then the state space of the joint system is . Now suppose that Alice makes a measurement on** system , and Bob makes a measurement on system . These measurements are represented physically by unitary transformations and . The effect of the measurements on the joint system are therefore represented by the unitary transformations and , where and are the identity transformations on and , respectively. The key to the no-communication theorem is the observation that the transformations and commute with each other. (Either way you take the product you get .) 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 ; 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 and replaced by more realistic operators.

Comment author: Dacyn 04 August 2014 12:15:34AM 0 points [-]

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...

Comment author: Dacyn 03 August 2014 11:49:40PM *  0 points [-]

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 and , whose state spaces are represented by Hilbert spaces and , respectively. Then the state space of the joint system is . Now suppose that Alice makes a measurement on** system , and Bob makes a measurement on system . These measurements are represented physically by unitary transformations and . The effect of the measurements on the joint system are therefore represented by the unitary transformations and , where and are the identity transformations on and , respectively. The key to the no-communication theorem is the observation that the transformations and commute with each other. (Either way you take the product you get .) 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 ; 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 and replaced by more realistic operators.

In response to The Quantum Arena
Comment author: Eliezer_Yudkowsky 16 April 2008 06:05:27PM 2 points [-]

Komponisto, I looked up your references and found that the Hilbert space of QM is generally believed to have a countable basis, though there are occasional theories which make the basis uncountable.

I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable. But in QM it's countable... because everything comes in multiples of Planck's constant, perhaps? Though I haven't seen the actual reason stated, and perhaps it's something beyond my current grasp.

In any case, I've edited the text to read simply "infinite" and not "uncountable".

When I was a kid, I learned that aleph-one was the cardinality of the set of reals, and in my heart, aleph-one will always be the cardinality of the set of reals. I'd say that I was assuming the continuum hypothesis, except that I'm an infinite set atheist.

Comment author: Dacyn 03 August 2014 11:02:25PM *  6 points [-]

I'd thought the Hilbert space was uncountably dimensional because the number of functions of a real line is uncountable. But in QM it's countable... because everything comes in multiples of Planck's constant, perhaps? Though I haven't seen the actual reason stated, and perhaps it's something beyond my current grasp.

Ahh... here's something I can help with. To see why Hilbert space has a countable basis, let's first define Hilbert space. So let

= the set of all functions such that the integral of is finite, and let

= the set of all functions such that the integral of is zero. This includes for example the Dirichlet function which is one on rational numbers but zero on irrational numbers. So it's actually a pretty big space.

Hilbert space is defined to be the quotient space . To see that it has a countable basis, it suffices to show that it contains a countable dense set. Then the Gram-Schmidt orthogonalization process can turn that set into a basis. What does it mean to say that a set is dense? Well, the metric on Hilbert space is given by the formula

,

so a sequence is dense if for every element of Hilbert space, you can find a sequence such that . Now we can see why we needed to mod out by -- any two points of are considered to have distance zero from each other!

So what's a countable dense sequence? One sequence that works is the sequence of all piecewise-linear continuous functions with finitely many pieces whose vertices are rational numbers. This class includes for example the function defined by the following equations:

for all

for all

for all

for all

Note that I don't need to specify what does if I plug in a number in the finite set , since any function which is zero outside of that set is an element of , so would represent the same element of Hilbert space as .

So to summarize:

  1. The uncountable set that you would intuitively think is a basis for Hilbert space, namely the set of functions which are zero except at a single value where they are one, is in fact not even a sequence of distinct elements of Hilbert space, since all these functions are elements of , and are therefore considered to be equivalent to the zero function.

  2. The actual countable basis for Hilbert space will look much different, and the Gram-Schmidt process I alluded to above doesn't really let you say exactly what the basis looks like. For Hilbert space over the unit interval, there is a convenient way to get around this, namely Parseval's theorem, which states that the sequences and form a basis for Hilbert space. For Hilbert space over the entire real line, there are some known bases but they aren't as elegant, and in practice we rarely need an explicit countable basis.

  3. Finally, the philosophical aspect: Having a countable basis means that elements of Hilbert space can be approximated arbitrarily well by elements which take only a finite amount of information to describe*, much like real numbers can be approximated by rational numbers. This means that an infinite set atheist should be much more comfortable with countable-basis Hilbert space than with uncountable-basis Hilbert space, where such approximation is impossible.

* The general rule is:

Elements of a finite set require a finite and bounded amount of information to describe.

Elements of a countable set require a finite but unbounded amount of information to describe.

Elements of an uncountable set (of the cardinality of the continuum) require a countable amount of information to describe.

Comment author: HalFinney 24 April 2008 05:31:06AM 0 points [-]

The problem with locality and the position basis is that the Schrodinger equation doesn't fully enforce locality. With a single particle, it does, but with a multi-particle configuration, conditions near particle 1 can affect the evolution of a configuration that involves particle 2. Somehow these kinds of correlations and influences happen while still not enabling FTL communication, but I don't know of anything in the formalism that clearly enforces this limitation.

Comment author: Dacyn 03 August 2014 10:29:14PM 3 points [-]

FTL communication is not ruled out by the Schrodinger equation, but this is irrelevant because the Schrodinger equation is not valid for systems which include fast-moving particles. Instead, you have to use quantum field theory, of which the Schrodinger equation is the limit as the speed of light approaches infinity. In QFT, FTL communication is indeed ruled out by the formalism, as you suggest. Specifically, it's the commutativity or anticommutativity of field operators based at points which are spacelike separated that does it. For further details I would suggest reading the short paper of Eberhard and Ross. (Unfortunately you need an institutional affiliation to view the link, but I can send a PDF to anyone who wants it.)

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