Experimental estimation of the dimension of classical and quantum systems
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 behavior of the system under study. The dimension of this system, that is, the number of relevant and independent parameters 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, since 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. That is, is the standard initial assumption on the dimension unavoidable? If not, what can be said about the dimension of an unknown system only from the observed measurement data and without making any assumption about the detailed functioning of the devices used in the experiment? The concept of a dimension witness answers this question, as it allows bounding the dimension of an unknown system only from measurement statistics. In a recent work, we report the first experimental demonstration of a dimension witnesses. We use photon pairs entangled in polarization and orbital angular momentum to generate ensembles of classical and quantum states of dimensions up to four. We then use a dimension witness to certify their dimensionality as well as their quantum nature. Proving that the dimension of an unknown system is an experimentally accessible quantity is a fundamental result. Besides its fundamental interest, our work opens new avenues in quantum information science, where dimension represents a powerful resource, especially for device-independent estimation of quantum systems and quantum communications.