Fundamental limits on quantum cloning from the no-signalling principle

The no-cloning theorem states that within the framework of quantum mechanics, there does not exist any universal procedure that can replicate an unknown quantum state reliably. This fundamental principle was initially formalized in 1982 and later extended in various directions. The no-cloning theorem has rich implications in quantum cryptography, ensuring the security of primitives such as quantum money, quantum key distribution and quantum secret sharing.

However, quantum mechanics still has unsolved problems and may need further refinement. It is essential to question whether the no-cloning theorem and consequentially the security of quantum cryptographic primitives still hold even if quantum mechanics is modified. The no-signalling principle states that information cannot be transmitted faster than light. It is believed to be a fundamental physical principle which will remain solid even if quantum mechanics is replaced by a more refined theory. It has been noticed that the no-cloning theorem remains true in different situations as long as the no-signalling principle is respected. However, these works are restricted to specific situations.

We presents a general scheme to obtain bounds on the worst-case fidelity of any probabilistic or deterministic cloning machine from the no-signalling principle. Here we assume that the predictions of quantum mechanics are correct so that we can, in particular, perform a remote state preparation protocol, but we do not assume that the cloning machine itself adheres to the laws of quantum mechanics. Thus, we can restrict to cloning machines that are not necessarily linear in their input and are not necessarily positive or trace-preserving on general mixed input states.

We introduce a novel protocol for remote identical state preparation, enabling Alice to remotely prepare n copies of a state drawn probabilistically from a selected set for Bob. Importantly, both Alice and Bob can independently verify the success of the state preparation without the need for communication. By employing a contradiction-based argument, we establish the impossibility of a cloning machine. Consider the scenario where Alice and Bob engage in the remote identical state preparation protocol, and Bob attempts to duplicate the output into m copies using a cloning machine. If Bob were able to generate m copies of the state with high accuracy, he could deduce which set Alice had selected. Consequently, Alice could effectively signal this information to Bob, resulting in a violation of the no-signalling principle. Hence, the no-signalling principle acts as a fundamental constraint preventing the existence of a cloning machine.

Our general scheme is versatile and can be employed to study fundamental limits for cloning machines that only attempt to clone various specific subsets of states and unitary gates, including general states, spin-coherent states and stabilizer states. Moreover, the no-cloning bounds we derive from the no-signalling principle can sometimes reproduce the strongest no-cloning bounds that are derived assuming that the cloning machines are obeying the rules of quantum mechanics.

Interested readers may find the full version in ArXiv.

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