A fundamental resource in any communication and computation task is the amount of information that can be transmitted and processed. The classical information encoded in a set of states is limited by the number of distinguishable states or classical dimension d c of the set. The sets used in quantum communication and information processing contain states that are neither identical nor distinguishable, and the quantum dimension d q of the set is the dimension of the Hilbert space spanned by these states. An important challenge is to assess the (classical or quantum) dimension of a set of states in a device-independent way, that is, without referring to the internal working of the device generating the states. Here we experimentally test dimension witnesses designed to efficiently determine the minimum dimension of sets of (three or four) photonic states from the correlations originated from measurements on them, and distinguish between classical and quantum sets of states.Classical and quantum dimensions are fundamental quantities in information processing. In particular, the security of many cryptographic schemes 1-3 crucially relies on the dimensional characteristics of the information carriers. From a fundamental perspective, the difference between classical and quantum dimensions can be used for quantification of the non-classicality of correlations: classical simulation of correlations produced by a quantum system of (quantum) dimension d q may require a classical system of (classical) dimension d c d q (refs 4-6).The problem of efficiently testing the minimum possible dimension spanned by a set of states has been approached from different theoretical perspectives. The concept of a quantum dimension witness was first introduced for the dimension of the Hilbert space of composite systems tested locally 7 , and then related to the construction of quantum random access codes 8 and approached from a dynamical viewpoint 9 .A device-independent approach, that is, without any reference to the internal working of the device generating the states (state preparator) was introduced recently 10 . In this scenario, the measurement device must be trusted. Such trust can be based on, for example, the device successfully passing suitable tests before the test of the state preparator. Moreover, one has to assume that the manufacturers of the state preparator and the measurement device do not conspire against the user. This implies that there are no secret communication channels or preprogrammed correlations between the state preparator and the measurement device.The state preparation and the tests performed under these premises are shown schematically in Fig. 1. There is a state preparator with N buttons; it emits a particle in a state ρ x (specified by the device's supplier) when button x ∈ {1,...,N } is pressed. For testing, the emitted particles are sent to a measurement 1 Physics Department, Stockholm University, S-10691 Stockholm, Sweden, 2 Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spa...

We report two fundamental experiments on three-level quantum systems (qutrits). The first one tests the simplest task for which quantum mechanics provides an advantage with respect to classical physics. The quantum advantage is certified by the violation of Wright's inequality, the simplest classical inequality violated by quantum mechanics. In the second experiment, we obtain contextual correlations by sequentially measuring pairs of compatible observables on a qutrit, and show the violation of Klyachko et al.'s inequality, the most fundamental noncontextuality inequality violated by qutrits. Our experiment tests exactly Klyachko et al.'s inequality, uses the same measurement procedure for each observable in every context, and implements the sequential measurements in any possible order.

A postselection step in quantum cryptography based on energy-time entanglement makes the system insecure.

Contextuality is a fundamental property of quantum theory and a critical resource for quantum computation. Here, we experimentally observe the arguably cleanest form of contextuality in quantum theory [A. Cabello et al., Phys. Rev. Lett. 111, 180404 (2013)] by implementing a novel method for performing two sequential measurements on heralded photons. This method opens the door to a variety of fundamental experiments and applications.

We report on an experimental test of classical and quantum dimension. We have used a dimension witness that can distinguish between quantum and classical systems of dimensions two, three, and four and performed the experiment for all five cases. The witness we have chosen is a base of semi-device-independent cryptographic and randomness expansion protocols. Therefore, the part of the experiment in which qubits were used is a realization of these protocols. In our work we also present an analytic method for finding the maximum quantum value of the witness along with corresponding measurements and preparations. This method is quite general and can be applied to any linear dimension witness.