It has been shown that the conditional probability distributions obtained by performing measurements on an uncharacterized physical system can be used to infer its underlying dimension in a device-independent way both in the classical and the quantum setting. We analyze several aspects of the structure of the sets of probability distributions corresponding to a certain dimension, taking into account whether shared randomness is available as a resource. We first consider the so-called prepare-and-measure scenario. We show that quantumness and shared randomness are not comparable resources. That is, on the one hand there exist behaviors that require a quantum system of arbitrarily large dimension in order to be observed while they can be reproduced with a classical physical system of minimal dimension together with shared randomness. On the other hand, there exist behaviors that require exponentially larger dimensions classically than quantumly even if the former is supplemented with shared randomness. We also show that in the absence of shared randomness, the sets corresponding to a sufficiently small dimension are negligible (zero measure and nowhere dense) both classically and quantumly. This is in sharp contrast to the situation in which this resource is available, and it explains the exceptional robustness of dimension witnesses in the setting in which devices can be taken to be uncorrelated. We finally consider the Bell scenario in the absence of shared randomness, and we prove some nonconvexity and negligibility properties of these sets for sufficiently small dimensions. This shows again the enormous difference induced by the availability (or lack thereof) of this resource.
It has been shown that the conditional probability distributions obtained by performing measurements on an uncharacterized physical system can be used to infer its underlying dimension in a device-independent way both in the classical and the quantum setting. We analyze several aspects of the structure of the sets of probability distributions corresponding to a certain dimension, taking into account whether shared randomness is available as a resource. We first consider the so-called prepare-and-measure scenario. We show that quantumness and shared randomness are not comparable resources. That is, on the one hand there exist behaviors that require a quantum system of arbitrarily large dimension in order to be observed while they can be reproduced with a classical physical system of minimal dimension together with shared randomness. On the other hand, there exist behaviors that require exponentially larger dimensions classically than quantumly even if the former is supplemented with shared randomness. We also show that in the absence of shared randomness, the sets corresponding to a sufficiently small dimension are negligible (zero measure and nowhere dense) both classically and quantumly. This is in sharp contrast to the situation in which this resource is available, and it explains the exceptional robustness of dimension witnesses in the setting in which devices can be taken to be uncorrelated. We finally consider the Bell scenario in the absence of shared randomness, and we prove some nonconvexity and negligibility properties of these sets for sufficiently small dimensions. This shows again the enormous difference induced by the availability (or lack thereof) of this resource.
Device-independent dimension witnesses provide a lower bound on the dimensionality of classical and quantum systems in a ``black box''scenario where only correlations between preparations, measurements, and outcomes are considered. We address the problem of the robustness of dimension witnesses, namely that to witness the dimension of a system or to discriminate between its quantum or classical nature, even in the presence of loss. We consider the case when shared randomness is allowed between preparations and measurements and provide a threshold in the detection efficiency such that dimension witnessing can still be performed.
Abstract The prepare-and-measure scenario offers the possibility to infer the dimension of an unknown physical system in a device-independent way, i.e. using only raw measurement data with apparatuses regarded as black boxes. We provide here a general lower bound on the dimension necessary to observe arbitrary quantum behaviours in this scenario based on simple matrix analysis. This bound holds even if the preparer and measurer share randomness. This is relevant in scenarios where the parties are free to access this resource or it is not safe to assume that the devices are not correlated. We further use this result to bound the success probability of random access codes in general as a function of the dimension of the quantum systems sent from one party to another and we provide constructions of dimension witnesses.
The device-independent formalism is a set of tools to analyze experimental data and infer properties about systems, while avoiding almost any assumption about the functioning of devices. It has found applications both in fundamental and applied physics: some examples are the characterization of quantum nonlocality and information protocols for secure cryptography or randomness generation. This thesis contains novel results on these topics and also new applications such as deviceindependent test for dimensionality. After an introduction to the field, the thesis is divided in four parts. In the first we study device-independent tests for classical and quantum dimensionality. We investigate a scenario with a source and a measurement device. The goal is to infer, solely from the measurement statistics, the dimensionality required to describe the system. To this end, we exploit the concept of dimension witnesses. These are functions of the measurement statistics whose value allows one to bound the dimension. We study also the robustness of our tests in more realistic experimental situations, in which devices are affected by noise and losses. Lastly, we report on an experimental implementation of dimension witnesses. We conducted the experiment on photons manipulated in polarization and orbital angular momentum. This allowed us to generate ensembles of classical and quantum systems of dimension up to four. We then certified their dimension as well as its quantum nature by using dimension witnesses. The second part focuses on nonlocality. The local content is a nonlocality quantifier that represents the fraction of events that admit a local description. We focus on systems that exhibit, in that sense, maximal nonlocality. By exploiting the link between Kochen-Specker theorems and nonlocality, we derive a systematic recipe to construct maximally nonlocal correlations. We report on the experimental implementation of correlations with a high degree on nonlocality in comparison with all previous experiments on nonlocality. We also study maximally nonlocal correlations in the multipartite setting, and show that the so-called GHZ-state can be used to obtain correlations suitable for multipartite information protocols, such as secret-sharing. The third part studies nonlocality from an operational perspective. We study the set of operations that do not create nonlocality and characterize nonlocality as a resource theory. Our framework is consistent with the canonical definitions of nonlocality in the bipartite setting. However, we find that the well-established definition of multipartite nonlocality is inconsistent with the operational framework. We derive and analyze alternative definitions of multipartite nonlocality to recover consistency. Furthermore, the novel definitions of multipartite nonlocality allows us to analyze the validity of information principles to bound quantum correlations. We show that `information causality' and `non-trivial communication complexity' are insufficient to characterize the set of quantum correlations. In the fourth part we present the first quantum protocol attaining full randomness amplification. The protocol uses as input a source of imperfect random bits and produces full random bits by exploiting nonlocality. Randomness amplification is impossible in the classical regime and it was known to be possible with quantum system only if the initial source was almost fully random. Here, we prove that full randomness can indeed be certified using quantum non-locality under the minimal possible assumptions: the existence of a source of arbitrarily weak (but non-zero) randomness and the impossibility of instantaneous signaling. This implies that one is left with a strict dichotomic choice regarding randomness: either our world is fully deterministic or there exist events in nature that are fully random.
We introduce the dimension-restricted prepare and broadcast (PAB) scenario, which generalizes standard prepare-and-measure frameworks. Here, the system prepared by a sender undergoes a broadcasting transformation before being locally measured by multiple receivers. We develop a hierarchy of classical, quantum, and nonsignalling models describing this scenario, characterize their corresponding correlation sets, and derive new families of Bell-like inequalities together with linear and semidefinite programming methods for their certification. First, assuming shared randomness, we prove that the hierarchy collapses into a single set whenever we consider only one measurement per party. Then, considering multiple possible measurements, we show that PAB scenarios allow the activation of nonclassicality, revealing genuinely nonclassical features in resources that admit classical descriptions in standard prepare-and-measure or Bell settings.
Measurement incompatibility stipulates the existence of quantum measurements that cannot be carried out simultaneously on single systems. We show that the set of input-output probabilities obtained from d-dimensional classical systems assisted with shared randomness is the same as the set obtained from d-dimensional quantum strategies restricted to compatible measurements with shared randomness in any communication scenario. Thus, measurement incompatibility is necessary for quantum advantage in communication, and any quantum advantage (with or without shared randomness) in communication acts as a witness to the incompatibility of the measurements at the receiver's end in a semi-device-independent way. We introduce a class of communication tasks---a general version of random access codes---to witness incompatibility of an arbitrary number of quantum measurements with arbitrary outcomes acting on d-dimensional systems and provide generic upper bounds on the success metric of these tasks for compatible measurements. We identify all sets of three incompatible rank-one projective qubit measurements that random access codes can witness. Finally, we present the generic relationship between different sets of probability distributions---classical, quantum with or without shared randomness, and quantum restricted to compatible measurements with or without shared randomness---produced in communication scenarios.
We investigate the prepare-and-broadcast scenario, a multipartite extension of dimension-constrained prepare-and-measure experiments in which a quantum system is distributed to multiple receivers. We derive fundamental trade-offs between prepare-and-measure witnesses, Bell nonlocality, and quantum random access code performance. We further develop a semi-device-independent randomness certification framework based on prepare-and-broadcast witnesses, showing that the maximal quantum violation certifies two bits of joint randomness, exceeding the limit achievable from the CHSH inequality while remaining robust to noise. Finally, we show that the prepare-and-broadcast scenario naturally accommodates stronger adversarial models in which the eavesdropper retains a quantum system correlated with the measurement device, providing a natural framework for semi-device-independent randomness certification against quantum side information.