Experimental estimation of the dimension of classical and quantum systems

Abstract

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...

Device-Independent Certification of High-Dimensional Quantum Systems

“…This reveals that assessing high dimensions in a DI manner constitutes an experimental challenge, since it requires much higher control and accuracy than previous experiments assessing dimensions in a DI manner [17,18]. This is the challenge that we address here.…”

“…So far, DI DWs have only been used to experimentally certify the generation of classical and quantum systems of dimension 2 and 3 [17,18]. However, realistic quantum information processing applications demand quantum systems of much higher dimensions.…”

Shared randomness and device-independent dimension witnessing

“…These results are not only interesting from the fundamental point of view but also play a role in quantum information processing. Besides allowing for experimental tests of the physical dimension [1], which might be considered as a resource, these investigations allow to constrain the correlations that are achievable when the setting limits the underlying dimension of the physical systems used in a protocol. These scenarios are know as semi-device-independent quantum information processing: no assumption is made on the working of the devices nor on the physical systems used except for its dimension.…”

Three-observer classical dimension witness violation with weak measurement

“…To estimate the lower bound of the dimensions of a physical system, quantum dimension witness has been proposed and experimentally realized [6][7][8][9] , which has important applications in the semi-device-independent quantum key distribution and quantum random number generation [10][11][12][13][14][15] . Until now, it has been demonstrated that two-observer classical dimension witness violation can be achieved with the Bell inequality test, quantum random access code test, and determinant value test respectively [16][17][18] , but whether the multi-observer classical dimension witness violation can be obtained or not is still an open question.…”