Contextual advantage for state-dependent cloning

Matteo Lostaglio1,2 and Gabriel Senno1

1ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
2QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands

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A number of noncontextual models exist which reproduce different subsets of quantum theory and admit a no-cloning theorem. Therefore, if one chooses noncontextuality as one's notion of classicality, no-cloning cannot be regarded as a nonclassical phenomenon. In this work, however, we show that there are aspects of the phenomenology of quantum state cloning which are indeed nonclassical according to this principle. Specifically, we focus on the task of state-dependent cloning and prove that the optimal cloning fidelity predicted by quantum theory cannot be explained by any noncontextual model. We derive a noise-robust noncontextuality inequality whose violation by quantum theory not only implies a quantum advantage for the task of state-dependent cloning relative to noncontextual models, but also provides an experimental witness of noncontextuality.

It has been known for 50 years now that quantum information, differently from classical information, cannot be cloned. However, as we have learned in the last years, this $\textit{a priori}$ genuine quantum feature can be perfectly emulated classically by imagining that, when we experimentally prepare a quantum state, what is "really" going on is that we are preparing one of many ontic (a.k.a. hidden) states according to some probability distribution. If overlapping quantum states correspond to overlapping probability distributions, then we obtain a simple explanation of the no-cloning theorem. So, one may wonder, is there anything truly nonclassical in the phenomenology of quantum cloning?

In this work, we studied this question in terms of how well quantum mechanics clones a given pair of (in general) overlapping states, using a set of fundamental results worked out during the hey-days of quantum information theory. We showed that the tradeoff between the fidelity of the input pair and that of the (necessarily) imperfect clones is too good in quantum theory to be explainable in terms of the overlap of probabilities over some classical states. Specifically, we showed that $contextuality$, already identified as a necessary resource for universal quantum computation and optimal state discrimination, is what allows quantum mechanics to clone better than one can anticipate from classical emulations. Our main contribution is a noise-robust noncontextuality inequality whose violation by quantum theory not only $\textit{implies a quantum advantage for the task of state-dependent cloning}$ but also provides an experimental witness of noncontextuality. We expect our methods to be suitable to identify nonclassicality in other quantum phenomena such as, e.g., quantum teleportation.

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