I'm familiar with the "no cloning" theorem (and its proof) which roughly states that no linear gate (circuit) can clone an unknown state (obviously by unitary transformations).

Lately I've heard about different, more fundamental limitation on cloning which is derived from entropy considerations.

My question is if there is really such a (different) limitation (and in that case I'd like to recieve a reference to a paper about it) or is it just a loose restatement of the original theorem, that is conservation of information in system which evolve unitarily.

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    $\begingroup$ Do you have a reference? "Entropy considerations" is kind of vague, as is "more fundamental limitations". Could be monogamy of entanglement, could be infinite classical information per transmitted qubit, could be a lot of things. $\endgroup$ Oct 4 '16 at 3:13
  • $\begingroup$ If I had more details I could search it on my own. I ask here because of the experience of people who may done some work in this field and may recognize what I'm talking about. $\endgroup$
    – Alexander
    Oct 4 '16 at 3:16
  • $\begingroup$ Well what context did you hear it in? Papers about optimal approximate cloning might mention it in passing, e.g. arxiv.org/abs/quant-ph/9804001 $\endgroup$ Oct 4 '16 at 3:17
  • $\begingroup$ So would you be happy with any answer involving an entropic quantity (mutual information etc.)? $\endgroup$ Oct 6 '16 at 21:23
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    $\begingroup$ You could find this paper interesting. $\endgroup$
    – valerio
    Oct 12 '16 at 19:42

The basic idea of no-cloning is that, under unitary evolution you can't copy an unknown state. I don't precisely know what entropy related no-cloning theorem you are referring to. However, one uses no-cloning arguments for quantum capacity problems, which involve entropy terms.

Here is an example of that thinking: The capacity of a qauntum channel is characterized by an entropy difference (for basic definitions see 'Quantum Channel Capacities' by Graeme Smith (2010)). Using a no-cloning 'type' argument, for some channels this capacity must be zero, which gives a bound on the entropy difference. One place this no-cloning type argument is invoked is 'Capacities of Quantum Erasure Channels' by Bennett et al (1997). Here is how that works:

A quantum channel, has two outlets the output and the environment. Say, a channel for any input, sends exactly the same state to the output and the environment (this is possible). Then such a channel cannot have any quantum capacity. If it did, then over many uses of the channel, some set of unknown quantum states can be sent perfectly from input to output (by definition of quantum capacity). The same state can also be received at the environment (since output and environment receive exactly the same state). This would mean, we have copied an unknown input into two. This violates our regular no cloning, since the whole system (input = [output + environment]) evolves unitarily.

  • $\begingroup$ It would be preferable if you could provide the essence of the discussion in the references, which you have given in your answer. $\endgroup$
    – Gonenc
    Oct 12 '16 at 16:55
  • $\begingroup$ The essence of the discussions in the references pertaining to the original question has been edited into the previous answer. The argument, which I have written is my understanding of the reasons given in the Bennett et al paper. $\endgroup$
    – Vikesh
    Oct 12 '16 at 19:36

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