Can the requirement of a quantum code to be error correcting be expressed in terms of a relation involving only entropic quantities (von neumann entropy, mutual information etc)?

For example, a lot of attention has been given to the study of reconstruction of quantum states (see for example, http://arxiv.org/abs/1410.0664) , which is related to conditional mutual information, and sounds very similar to the requirements of quantum error correction.

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    $\begingroup$ Could you elaborate what you call "quantum code" and what you call "quantum information"? For me, quantum information is about sending and processing information with quantum theory. This includes quantum Shannon theory, but also quantum error-correction and quantum cryptography. $\endgroup$ – Martin Dec 23 '14 at 12:07
  • $\begingroup$ Sure. I was quite vague in the way I asked the question. Hope its much clearer now. $\endgroup$ – anurag anshu Dec 25 '14 at 14:50

There is a very easy, but highly unsatisfying way that your question could be answered:

Given a noisy channel, a code is error-correcting, iff it increases the capacity. Normally, the idea is that the channel has zero-capacity and afterwards, it has capacity.

Now, the various capacities (e.g. classical, quantum-classical, quantum, one-shot-quantum) are usually expressed via entropic quantities (e.g. Shannon's noisy channel theorem, the LSD-theorem), which seems to answer the question.

However, obviously, the requirement of a quantum channel to be error-correcting, if phrased this way, is very broad and usually very hard to deal with. In essence, the whole point of the subfield of error-correction is to find better ways to express this. I do have some ideas of where to look for more connections between entropy and error-correcting abilities, but I have only ever heard them floating around and I am more on the quantum Shannon than the error-correcting side of QI. One principle would be that a noisy channel typically increases the entropy of the input state, i.e. to obtain error-correcting-codes, one would have to "get rid" of the additional entropy by concentrating them in some redundant part of the code. But I have never seen this fleshed out a lot. There is a paper, which might use this concept, but I haven't really looked at it and I don't know the author, so I can't say anything about it: http://arxiv.org/abs/quant-ph/0703258.


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