Pusey et al. reply —
We did not claim in our original Article (M. F. Pusey, J. Barrett & T. Rudolph, Nature Phys. 8, 475–478; 2012) that all models of the form considered in which the quantum state is not real can be ruled out by a single experiment. Rather, we claimed that for any such model there exists an experiment that is, in principle, feasible and would rule it out. The argument presented by Halataei (S. M. H. Halataei Nature Phys. 10, 174; 2014) “limits the conclusion one can draw from even a successful test” no further than this. We agree with the mathematical claim of the comment: for a fixed value of ε, the experimental noise, an infinite number of quantum state pairs can always be found that are sufficiently close together that an experiment with ε noise cannot place a non-trivial lower bound on the classical trace distance D. The conclusions of our paper rest on a different claim: for a fixed pair of quantum states it is always possible to choose a value of ε small enough so that D can be bounded arbitrarily close to 1, as shown in eq. (7) of our Article. There is — according to standard quantum theory — no fundamental limit on how small ε can be in a careful enough experiment. Hence, any model with overlapping probability distributions for a distinct pair of quantum states makes different predictions from standard quantum theory, and such a model could be ruled out in a suitably sensitive experiment. Difficulties with the finite precision of experiments are unavoidable in physics. There will always be an infinite number of theories that predict the same results to within current experimental error. There is always the hope that future experiments will be able to distinguish them, and in the meantime we can use other criteria, such as simplicity, to select a preferred theory. We join with most physicists in preferring quantum theory at present.
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