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Experimental observations are usually described using theoretical models that make assumptions about the dimensionality of the system under consideration. However, would it be possible to assess the dimension of a completely unknown system only from the results of measurements performed on it, without any extra assumption? The concept of a dimension witness 1-6 answers this question, as it allows bounding the dimension of an unknown system only from measurement statistics. Here, we report on the experimental demonstration of dimension witnesses in a prepare and measure scenario 6 . We use photon pairs entangled in polarization and orbital angular momentum 7-9 to generate ensembles of classical and quantum states of dimensions up to 4. We then use a dimension witness to certify their dimensionality as well as their quantum nature. Our work opens new avenues in quantum information science, where dimension represents a powerful resource 10-12 , especially for device-independent estimation of quantum systems 13-16 and quantum communications 17,18 .Dimensionality is one of the most basic and essential concepts in science, inherent to any theory aiming at explaining and predicting experimental observations. In building up a theoretical model, one makes some general and plausible assumptions about the nature and the behaviour of the system under study. The dimension of this system, that is, the number of relevant and independent degrees of freedom needed to describe it, represents one of these initial assumptions. In general, the failure of a theoretical model in predicting experimental data does not necessarily imply that the assumption on the dimensionality is incorrect, because there might exist a different model assuming the same dimension that is able to reproduce the observed data.A natural question is whether this approach can be reversed and whether the dimension of an unknown system, classical or quantum, can be estimated experimentally. Clearly, the best one can hope for is to provide lower bounds on this unknown dimension. Indeed, every physical system has potentially an infinite number of degrees of freedom, and one can never exclude that they are all necessary to describe the system in more complex experimental arrangements. The goal, then, is to obtain a lower bound on the dimension of the unknown system from the observed measurement data without making any assumption about the detailed functioning of the devices used in the experiment. Besides its fundamental interest, estimating the dimension of an unknown quantum system is also relevant from the perspective of quantum information science, where the Hilbert space dimension is considered as a resource. For instance, using higher-dimensional Hilbert spaces simplifies quantum logic for quantum computation 10 , enables the optimal realization of information-theoretic protocols 11,19,20 and allows for lower detection efficiencies in Bell experiments 21,22 . Moreover, the dimension of quantum systems plays a crucial role in security proofs of standard quantum...

A secure quantum identification system combining a classical identification procedure and quantum key distribution is proposed. Each identification sequence is always used just once and new sequences are "refuelled" from a shared provably secret key transferred through the quantum channel. Two identification protocols are devised. The first protocol can be applied when legitimate users have an unjammable public channel at their disposal. The deception probability is derived for the case of a noisy quantum channel. The second protocol employs unconditionally secure authentication of information sent over the public channel, and thus it can be applied even in the case when an adversary is allowed to modify public communications. An experimental realization of a quantum identification system is described. PACS 03.67D

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