# Classical and quantum limits to classical copying?

The no-cloning theorem can be sharpened to give quantitative bounds on the fidelity with which an arbitrary quantum state can be copied. Is there a similar picture available for classical copying? This breaks down into two

Questions:

1. In classical physics, do there exist limitations on copying? For instance, I might speculate that it's possible to make copies of a classical state down to any desired resolution, but guess that some minimum amount of energy $$E(n)$$ is required to copy $$n$$ bits of information, and that $$E(n)$$ diverges as $$n \to \infty$$ (probably $$E(n)$$ is also dependent on other parameters such as temperature). Such a limit would presumably be closely related to Landauer's Principle.

2. An actual classical system is just an approximation of a quantum system. So it seems natural to speculate that there's some way to quantify "how far a system is from being classical", and bound the fidelity with which the system can copy information based on this measure. What is the key parameter here? Does it have to do with how big the Hilbert space for the "environment" is? Or with measures of decoherence?

A guess:

Part of the problem is that I don't actually know how to formalize the notion of "classical information" in a quantum system. But here's a shot at something like (2). Define a classical copier to be an operator $$H$$ on a Hilbert space $$V \otimes V$$ such that $$H(e_i \otimes v) = e_i \otimes e_i$$ for some specified basis $$e_i$$ and $$v$$ in $$V$$. One operates $$H$$ by preparing the system so that the input is known to be collapsed to some $$e_i \otimes v$$ before applying $$H$$ (by measuring some $$H'$$). Then it seems like one would quantify the limits of $$H$$ as a copier in terms of how precisely the eigenbasis of $$H$$ actually lines up with the eigenbasis of $$H'$$ (since in practice it's impossible to make them match up perfectly) and then applying the usual quantitative version of the no-cloning theorem.