We study the nonlocality of arbitrary dimensional bipartite quantum states. By computing the maximal violation of a set of multi-setting Bell inequalities, an analytical and computable lower bound has been derived for general two-qubit states. This bound gives the necessary condition that a two-qubit state admits no local hidden variable models. The lower bound is shown to be better than that from the CHSH inequality in judging the nonlocality of some quantum states. The results are generalized to the case of high dimensional quantum states, and a sufficient condition for detecting the non-locality has been presented.
Quantum nonlocality is a cornerstone property for the development of the socalled quantum protocols as well as for the characterisation, and possible generalisation of quantum theory. However, there are still fundamental aspects of this property that need to be addressed; in particular, its relation with the property of quantum entanglement. It is well known that every nonlocal state is an entangled one, yet there exist entangled local states. Therefore, within their original defnitions, these two properties are not equivalent. The study of this relationship has been put forward by means of the generalisation of nonlocality through the so-called activation of nonlocality scenarios. The idea is to only work with these entangled local states in order to try to recover (activate) nonlocality. In this Thesis, we have numerically addressed the properties of entanglement, nonlocality (mostly by means of the violation of the CHSH-inequality), as well as the following generalisations of nonlocality: k-copy nonlocality, hidden nonlocality, and the combination of these two scenarios, namely, hidden k-copy nonlocality and tensor/filter activation of hidden nonlocality. We have studied these properties for some bipartite entangled local states, and some two-qubit not CHSH-inequality violating states of interest. In particular, we have analysed these properties for two-qubit states that come from the study of quantum games' strategies and the dynamics of open quantum systems.
In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers
S1,S2≥1 of generalized quantum measurements at two sites. In the present article, we find analytically a new general condition sufficient for a nonseparable Werner state with a dimension
d≤min{S1,S2} to satisfy all Bell inequalities under every
S1×S2-setting correlation scenario with outcomes of an arbitrary spectral type, discrete or continuous
− that is, to be
S1×S2-setting Bell local, for short. For a variety of measurement settings
S1,S2, this new general bound is beyond (i) Werner's and Barrett's locality bounds for a nonseparable Werner state and (ii) the related result by Johnson and Viola [Phys. Rev. A 88, 032323 (2013)] on "sharability" of this state. When specialized to
d=2 and
min{S1,S2}=2,3, our new general bound improves upon the corresponding results by Terhal et. al. in [Phys. Rev. Lett. 90, 157903 (2003)], moreover, for
d=min{S1,S2}=2, it reduces exactly to the bound derived for this case via the CHSH inequality. We prove also explicitly in operator terms the result of Terhal et. el. that every nonseparable Werner state with a dimension
d>min{S1,S2} is
S1×S2-setting Bell local. The new results presented in this article are important for both the theory of Bell nonlocality and quantum applications based on this fundamental concept.
In this paper we study the nonlocal properties of two-qubit Werner states parametrized by the visibility parameter $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. A family of Bell inequalities is constructed that proves the two-qubit Werner states to be nonlocal for the parameter range $0.7056<p\ensuremath{\leqslant}1$. This is slightly wider than the range $0.7071<p\ensuremath{\leqslant}1$, corresponding to the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. This answers a question posed by Gisin in the positive, i.e., there exist Bell inequalities which are more efficient than the CHSH inequality in the sense that they are violated by a wider range of two-qubit Werner states.
The nonlocality of quantum states on a bipartite system is tested by comparing probabilistic outcomes of two local observables of different subsystems. For a fixed observable A of the subsystem its optimal approximate double A ' of the other system is defined such that the probabilistic outcomes of A ' are almost similar to those of the fixed observable A . The case of σ-finite standard von Neumann algebras is considered and the optimal approximate double A ' of an observable A is explicitly determined. The connection between optimal approximate doubles and quantum correlations is explained. Inspired by quantum states with perfect correlation, like Einstein–Podolsky–Rosen states and Bohm states, the nonlocality power of an observable A for general quantum states is defined as the similarity that the outcomes of A look like the properties of the subsystem corresponding to A '. As an application of optimal approximate doubles, maximal Bell correlation of a pure entangled state on is found explicitly.
This Master's thesis has two central subjects: the simulation of correlations generated by local measurements on entangled quantum states by local hidden-variables models and the revelation of hidden nonlocality. We present and detail the Werner's local model and the hidden nonlocality of some Werner states of dimension
d≥5, the Gisin-Degorre's local model for a Werner state of dimension
d=2 and the local model of Hirsch et al. for mixtures of the singlet state and noise, all of them for projective measurements. Finally, we introduce the local model for POVMs of Hirsch et al. for a state constructed upon the singlet with noise, that still violates the CHSH inequality after local filters are applied, hence presenting the so-called genuine hidden nonlocality.