# 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.

You seem to be mixing two very different concepts here. Quantum cloning is talking about the absolute limits of what is theoretically possible in a perfect world. In this absolute theoretical limit, yes we can derive how well quantum cloning can work, and we also know that classical cloning is nominally perfect.

There is then a separate question of how well you can clone once you add in realistic sources of imperfection. Things like imperfect alignment of bases, decoherence .... These are certainly things you could incorporate into a quantum description, although I'm not aware of whether anyone ever has. And you could take it further and look at the classical limit of that. However, once you hit the actual classical limit, remember that there essentially is a "preferred" basis. You don't run into issues of misalignment because there only is one basis. And because you've implicitly got cloning built in, you've got many, many copies of data, rendering error correction incredibly effective, so decoherence is almost a non-issue as well. After all, if you copy a file one your computer, you get an exact copy, don't you?

Yes, there are limits. The higher density try and store and manipulate data, the closer to single information carrying entities you get (e.g. small number of photons in a fibre optic cable), and you're closer to using quantum things. But that's just because we're greedy. You don't have to approach those limits, and we can choose to do things supremely robustly.

I would also say that Landauer's principle is a bit misleading. It says that if you're performing an irreversible operation, and if you do it at a particular temperature, that has certain implications. However, computations don't need to be done irreversibly. They can be done reversibly, and then you're operating outside the regime that Landauer's principle can talk about.

• Thanks! I think I could be persuaded that probably misalignment of bases is not the most relevant parameter. But it still seems to me there ought to be some parameter measuring how "classical" a system is. In the introduction to this article it's suggestively pointed out that the error tolerance of modern fiber optic communications is almost quantum-limited. Also, in what sense is "known" that classical copying is perfect? Landauer's principle contradicts the idea that classical erasure is perfect, and copying is not so different from erasure. – Tim Campion Jul 1 '19 at 16:27
• To clarify: my point regarding fiber optic communications was to challenge the notion that my computer copies files perfectly -- there are always sources of error, and some of them are fundamental quantum ones. Of course, fiber optic communications are different from personal computers, but perhaps not so relevantly different. – Tim Campion Jul 1 '19 at 16:29