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What are the current most important theoretical problems on quantum entanglement? What is that we don't yet understand about how it works? (Not considering interpretation etc problems)

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Why is this question a community wiki ? –  Frédéric Grosshans Jan 25 '11 at 16:17
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I think this is usual when one expects a list of examples instead of a specific answer to a specific question. –  Anthony Leverrier Jan 25 '11 at 16:26
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I see two current main axis of research around entanglement. I have not really followed the last years of research, so I don't know exactly which questions are open and which have been solved.

I sadly have no time to elaborate this week, but since its a community wiki, others can complete my answer.

Entanglement characterization

The question is on how to characterize entanglement as a resource and how to interconvert various entanglement form into others. While the problem has essentially been solved for two-party-entanglement, many open question subsists in n-party-entanglement.

For a recent technical review, you can read the Horodecki family paper in Review of Modern Physics (restricted access) (arXiv version).

Finding quantum physics from information theoretic consideration

It is well known that quantum entanglement does not allow any faster than light communication. This is called the "no-signalling" principle. However, the no-signalling principle does not forbids theories with stronger correlations than quantum mechanical entanglement (see here.) It has been shown that if one could build such a "super-entangled" machine, one could perform various unintuitive communication-complexity related tasks (see here and here for technical papers about this, and here and here for blog posts about the second paper). The interesting open question is the following: can we find all the quantum-entanglement correlations (and the quantum mechanics formalism) from reasonable information-theoretic constraints similar to the no-signaling principle.

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quantum entanglement is a basic property of multi-body quantum mechanics - or any quantum mechanics with at least two degrees of freedom - that has been understood from the mid 1920s. So for 85 years, there have been no open theoretical problems related to quantum entanglement and it has been understood that it works as well as how any doable experiment involving entanglement works even though the modern terminology related to quantum information has only been used for a few decades.

Sincerely Luboš

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I beg to differ on your sentence on"for 85 years, there have been no open theoretical problems related to quantum entanglement" . There are open problems in multi-partite entanglement, entanglement witnesses, and bound entanglement, to name a few problems which are not only terminology related. –  Frédéric Grosshans Jan 25 '11 at 15:35
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I didn't write there were no problems. I wrote that there were no theoretical problems. There are of course engineering problems - to sustain entanglement for a lot of time (to get quantum computers) and to solve particular algorithmic and other problems to make the machinery more useful. But these are applied physics and information technology problems, not theoretical problems. –  Luboš Motl Jan 25 '11 at 17:47
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@Lubos I'm sure you know the weight your words have. You can see this from the first comment where @Eelvex says: "Is there a point asking the same question for problems related to quantum entanglement?" What do you mean by a "theoretical problem"? Newton's laws of motion were put forward in the 17th century. Yet mechanics was in fervent till the 19th century. Entanglement is a notion. Our understanding of this notion evolves only when we consider problems involving it. It is incorrect to say that there are no theoretical problems involving entanglement. –  user346 Jan 25 '11 at 18:43
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@Luboš : According to your definition, there have been not theoretical problem in classical mechanics avter Newton, and all the development of the notion of energy, action and Hamiltonian mechanics in 19th century are maths without any theory ? –  Frédéric Grosshans Jan 26 '11 at 9:41
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@Frédéric: of course, the statement you attribute to me - even though I didn't write it - is true. The development of the Hamiltonian and Lagrangian mechanics wasn't a solution to any "open problem" in classical physics - because there hadn't been any. They were just new formalisms to describe the same physics, and they consequently offered no answers that would be impossible to obtain previously. From the viewpoint of actual problems and their solutions, the new notions of action, Hamiltonian etc. were just maths or formalisms without new physical theories. –  Luboš Motl Mar 30 '11 at 9:32
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