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I noticed for some quantum information processing protocols such as teleportation, entanglement distillation, bound entanglement activation, entanglement catalysis, the protocols seem to have a similar structure as:

(1) A certain task can not be achieved with a certain system configuration.

(2) Then an extra entangled system is attached to the original system.

(3) A local unitary operation is carried out (not always since in entanglement catalysis we do not do this)

(4) A local measurement is carried out.

(5) Communication of the measurement result.

(6) A local post-operation is applied depending on the measurement results.

(7) The task is accomplished (may need multiple iterations of the above operations)

For me, it seems that the key components are:

(1) Attaching an extra entangled system

(2) Local measurement

(3) Communication of measurement results and post-operations

What's the physical(mathematical) meaning of these operations? Why they all have such a structure? Is there anything in common among these protocols?

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If you just sent a qubit in some state $|\psi\rangle$ from one place to another, it would be very difficult to protect the information it instantiates from decoherence. The decoherence is a result of the dependence of an outside system on the quantum information in the qubit.

But by using entanglement it is possible for a system to instantiate quantum information that can't be read out of it by any measurement on that system alone - locally inaccessible information. This can happen because it is possible for the Heisenberg picture observables of a system to depend on some information even though the expectation values of the observables have no such dependence. For an explanation of this see

http://arxiv.org/abs/quant-ph/9906007

and

http://arxiv.org/abs/1109.6223.

So the pattern is that you entangle the qubit $Q_1$ whose information you want to transmit with another qubit $Q_2$. Each qubit then contains locally inaccessible information about the other that can be passed around more easily than the original quantum information. When you want access to the quantum information you do a measurement that involves measurement results from both systems.

More complicated schemes just involve composing operations of this kind in the right way:

http://arxiv.org/abs/quant-ph/0404132

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Entanglement is what cannot be created by local operations and classical communication (LOCC), so it can be used to do otherwise impossible tasks in a scenario where one is restricted to LOCC.

Now what is the most general LOCC protocol? A measures, communicates the outcome, B measures (conditional on the first outcome), communicates the outcome, A measures (conditional on all previous outcomes), etc.. This is a general multi-round LOCC protocol.

It just happens that many protocols can be carried out with just one LOCC round, which is exactly the steps you describe.

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  • $\begingroup$ I agree with your description. But for me this is more or less just superfacial. Yes, introducing an entangled subsystem does increase a kind of 'resource' for achieving a task. But how exactly this works? Personally I am not satisfied by playing with linear algebra and probabilities to explain it. I am expecting a more physically intuitive picture. $\endgroup$ – XXDD Aug 25 '15 at 10:57
  • $\begingroup$ @X.Dong I think the whole picture is very operational, and exactly not just linear algebra and probabilities. The operational aspect might have more of information theory and communication, put I think it is very pictorial. $\endgroup$ – Norbert Schuch Aug 25 '15 at 11:06
  • $\begingroup$ Still I agree with you that they can be understood from information/communication theory viewpoints and therefore have operational pictures. What I am interested is their geometrical picture. For me if the ER=EPR conjecture is true, then we must understand the geometry of entanglement and linear algebra/probability/communication theory is not preferred for this purpose, at least for me. $\endgroup$ – XXDD Aug 25 '15 at 11:35
  • $\begingroup$ @X.Dong This is not what you asked for. You keep asking some question and if one gives an answer, you ask something else, rather than accepting or rejecting the answer. I suspect this does not motivate people to answer your questions. $\endgroup$ – Norbert Schuch Aug 25 '15 at 11:52
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    $\begingroup$ @X.Dong: I'd advise to read the quick tour: physics.stackexchange.com/tour - then you know most you need to know to successfully interact at this site. Regarding the question, I can understand that you want a geometric picture - which is no more and probably LESS operational and maybe "physical" than the explanation given here - and there are people working on this, but it's hard. I don't know a single result that has a good "geometric" interpretation of LOCC operations... $\endgroup$ – Martin Aug 25 '15 at 13:09

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