# How can classical bits be copied if qubits cannot be copied?

The no-cloning theorem of quantum mechanics tells us there can be no general quantum circuit that can copy arbitrary qubit states, i.e. a quantum gate or circuit cannot send $$|0\rangle |\psi\rangle\mapsto|\psi\rangle |\psi\rangle$$ for arbitrary $$|\psi\rangle$$.

However, in classical circuits we can easily perform the copy $$(0, b)\mapsto(b, b)$$ by the function $$f:(a, b)\mapsto (a\oplus b, b)$$.

Now obviously, the no-cloning theorem doesn't apply because these are bits and not qubits, but the real question concerns the following dilemma. The above classical circuit can be physically realized, and the physical realization has to be ultimately reducible to quantum mechanics. If quantum mechanics truly explains our world, it must be able to give an account of the behavior of bits (not just qubits), and therefore it must be able to give an account of how bits can be cloned.

So I suppose my real question is, what is the quantum mechanical account of how classical bits can be copied?

There is a related question asking how is it possible that classical computations are non-unitary. According to the answer, unitarity only applies to isolated quantum systems and any open system (such as a system in which measurements are done) does not necessarily obey unitarity (this is similar to how open systems do not necessarily obey conservation of energy).

This gives me a partial answer, but it doesn't give me a deep enough understanding. For example, I can understand how measurements lead to the apparent destruction of information, and so I can see how a classical gate can collapse bits (e.g. see XOR-gate $$(a, b)\mapsto (a\oplus b)$$ or AND-gate $$(a, b)\mapsto (a\wedge b)$$). But I don't see how this could give an account of cloning bits.

How can this dilemma be resolved in the context of cloning bits?

• Good question. I think the keywords to search for are the “thermodynamics of computation”. Feb 15 at 3:28
• Roughly thinking, I think classical copy is only recreation, i.e., see it first, then create the same one. Then for qubits, by the nature of measurement, we may have the conclusion. Feb 15 at 5:21

TL;DR: The ban on copying is not nearly as universal as it might seem. No-cloning theorem actually allows copying as long as it is limited to orthogonal states. Classical information is the type of information which is encoded in orthogonal states, so it may be copied.

## Loophole in no-cloning theorem

No-cloning theorem says that there does not exist a unitary $$U$$ such that for all $$|\psi\rangle$$

$$U|0\rangle|\psi\rangle = |\psi\rangle|\psi\rangle.\tag1$$

The reason we can't have a $$U$$ like that is because we demand too much. No unitary can copy every possible $$|\psi\rangle$$. More precisely, no unitary can copy non-orthogonal states. However, if we instead settle for a $$U_\mathcal{B}$$ such that

$$U_\mathcal{B}|0\rangle|\psi\rangle=|\psi\rangle|\psi\rangle\tag2$$

for $$|\psi\rangle\in\mathcal{B}$$ where $$\mathcal{B}$$ is an orthonormal basis, then our demands turn out to be more realistic. Such $$U_\mathcal{B}$$ exists for every $$\mathcal{B}$$. For example, if $$\mathcal{B}=\{|0\rangle,|1\rangle\}$$ is the computational basis then the CNOT gate with the first qubit as the target and the second one as the control satisfies $$(2)$$.

See this earlier answer of mine for a little more details.

## Classical information

Classical information can be thought of as information encoded in orthonormal states. It has many properties characteristic to such states, e.g. it is possible to reliably distinguish classical states, it is possible to copy them and they are immune to decoherence.