This PhD-thesis is based on the five experiments I have performed during mytime as a PhD-student. Three experiments are implementations of non-contextualinequalities and two are implementations of witness functions for classical- andquantum dimensions of sets of states. A dimension witness is an operator function that produce a value whenapplied to a set of states. This value has different upper bounds depending onthe dimension of the set of states and also depending on if the states are classicalor quantum. Therefore a dimension witness can only give a lower bound on thedimension of the set of states.The first dimension witness is based on the CHSH-inequality and has theability of discriminating between classical and quantum sets of states of two andthree dimensions, it can also indicate if a set of states must be of dimension fouror higher.The second dimension witness is based on a set theoretical representationof the possible combinations of states and measurements and grows with thedimension of the set of states you want to be able to identify, on the other handthere is a formula for expanding it to arbitrary dimension.Non-contextual hidden variable models is a family of hidden variable modelswhich include local hidden variable models, so in a sence non-contextual inequal-ities are a generalisation of Bell-inequalities. The experiments presented in this thesis all use single particle quantum systems.The first experiment is a violation of the KCBS-inequality, this is the simplest correlation inequality which is violated by quantum mechanics.The second experiment is a violation of the Wright-inequality which is the simplest inequality violated by quantum mechanics, it contains only projectors and not correlations.The final experiment of the thesis is an implementation of a Hardy-like equality for non-contextuality, this means that the operators in the KCBS-inequality have been rotated so that one term in the sum will be zero for all non-contextual hidden variable models and we get a contradiction since quantum mechanicsgives a non-zero value for all terms.

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 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.

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.

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.