Testing the Dimension of Classical and Quantum Systems
Authors: Martin Hendrych1, Rodrigo Gallego1, Michal Mičuda1,2, Nicolas Brunner3, Antonio Acín1,4, Juan P. Torres1,5
1ICFO-Institut de Ciencies Fotoniques, Barcelona, Spain
2Department of Optics, Palacký University, Czech Republic
3H.H. Wills Physics Laboratory, University of Bristol, UK
4ICREA-Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain
5Department of Signal Theory and Communications, Universitat Politècnica de Catalunya, Barcelona, Spain
The main goal of any scientific theory is to predict and explain the results of experiments. In doing so, the theory makes some assumptions about the experiment under consideration, based, for instance, on some a priori knowledge, or symmetries of the setup. Based on these assumptions, a model -- possibly with some free parameters -- is constructed. The model is satisfactory whenever it is able to reproduce the observed results for reasonable values of the free parameters.
Consider for instance a quantum experiment involving measurements on several interacting particles. The quantum postulates tell us that the state, interactions and measurements in the setup should be described by operators acting on a Hilbert space of a given dimension. A standard practice is to assume that the dimension of the Hilbert space, that is, the number of independent parameters necessary to describe the setup is known. The theoretical model should then provide the operators in this space that reproduce the observed measurement statistics.
However one may ask whether this initial assumption on the dimension of the Hilbert space is in fact unavoidable to describe the experimental data or, on the contrary, if it is possible to estimate the dimension of a completely unknown quantum system only from the statistics of measurements performed on it. The concept of dimension witnesses gives a positive answer to the last question, as it provides lower bounds on the dimension of an unknown system only from the collected measurement results and without making any assumptions on the physical system under consideration. Clearly, without any assumption, the best one can hope for is to get lower bounds on the minimal dimension needed to describe an unknown system. In fact, one can never exclude the existence of further degrees of freedom in the system that are not seen in the present setup but can be accessed using a more refined experimental arrangement.
Our recent work  and a similar and independent experiment  represent the first experimental demonstrations of a dimension witness. Dimension witnesses were introduced in Reference in the context of Bell inequalities. Later, alternative techniques for bounding the dimension of unknown systems were proposed based on random access codes  or the time evolution of a quantum observable . In our experimental demonstration, we followed the approach presented in Reference, which applies to a “prepare and measurement” scenario as the one depicted in Figure 1. In this scenario there are two devices, the state preparator and the measurement device. These devices are seen as black boxes, as no assumptions are made on their internal working. At the state preparator, a quantum state ρx is prepared, out of N possible states. The state is then sent to the measuring device. There, a measurement y is performed, among M possible measurements, which produces a result b that can take K different values. The whole experiment is thus described by the probability distribution p(b|x,y), giving the probability of obtaining outcome b when measurement y is performed on the prepared state x. The goal is to estimate the minimal dimension of the mediating quantum particle between the two devices needed to describe the observed statistics.
Figure 1: Prepare-and-measure scenario for dimension witnesses. At the state preparator one can choose to prepare one out of N possible quantum states. The prepared state is denoted by ρx. The quantum state is then sent to the measuring device, where a measurement y is performed among M possibilities. The measurement result is denoted by b and can take K different values. For instance, the figure shows a scenario with four preparations and three measurements.
A dimension witness for a system of dimension d is simply a function of the observed probabilities that is bounded by a given value for all systems of dimension not larger than d. If in a given experiment the observed value of the dimension witness exceeds this bound, the system must necessarily have dimension larger than d. In our experiment, we observed the violation of one of the dimension witnesses introduced in , denoted by I4. The maximum values this witness can take for classical or quantum systems of dimension up to four are given in Table 1. Note that if the system dimension is assumed to be bounded, the witness also allows distinguishing between classical and quantum systems.
Table 1: Classical and quantum bounds for the dimension witness I4. The witness I4 can be used to discriminate ensembles of classical and quantum states of dimension up to 4. Note that for some values of the dimension a gap appears between classical and quantum systems. Thus, if one assumes a bound on the dimension of the system, the witness can be used to certify its quantum nature.
Obviously, to demonstrate the dimension witness, we need to construct quantum states of different dimensions. Fortunately, photons have a rich structure: they have polarization, frequency and spatial shape. Moreover, pairs of photons can be entangled . In our experiment we take advantage of all these features. First of all, multidimensional spaces of up to dimension 4 are created by generating photons in a superposition of two orthogonal polarization states (two dimensions) embedded into one out of two specific spatial modes (two more dimensions).
The state preparation works as follows. By means of spontaneous parametric down conversion, namely the generation of two lower frequency photons when a second order nonlinear crystal is pumped by an intense higher frequency optical beam, we generate photon pairs entangled in the polarization and spatial degrees of freedom. The detection of one of the photons in a tailored state effectively prepares (projects) the second photon in the desired quantum state. In our experiment, we are also interested in demonstrating the separation between quantum and classical systems of the same dimension. We achieve this at the preparation by exploiting the frequency degree of freedom, which is used to change the superposition that occurs in polarization from coherent (quantum) to incoherent (classical). The prepared photon is finally sent to the measuring device, where it is detected using optical tools very similar to those used in its generation: spatial light modulators, polarizers, optical fibers and single-photon counting modules.
Figure 2: (click on the image to view higher resolution version) Experimental results. The experiment probes the dimension witness I4 using systems of different nature, classical or quantum, and dimension (bit-qubit, trit-qutrit and quart). In the case of dimension 4 (quart), the dimension witness is insensitive to the quantum/classical transition (see also Table 1). For all dimensions, a violation of the corresponding bound is observed, certifying the dimension of the system.
To conclude, we have demonstrated that the dimension of classical and quantum systems can be bounded only from the measurement statistics without any extra assumption on the devices used in the experiment. Dimension witnesses represent an example of a device-independent estimation technique, in which relevant information about an unknown system is obtained only from the measurement data. Our work demonstrates how the device-independent approach can be employed to experimentally estimate the dimension of an unknown system. Beyond the fundamental motivation, the estimation of the dimension of unknown quantum systems is also relevant from a quantum information perspective, where the Hilbert space dimension is a resource that enables more powerful quantum information protocols. In fact, the quantum/classical distinction provided by dimension witnesses when the system dimension is bounded has recently been used for constructing protocols for secure key distribution  and randomness generation .
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