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Entanglement exhibits a unique property of quantum theory that is unseen in classical counterparts. The experimental confirmation of Bell’s inequalities has surprised many people and revived interest and debate regarding the exact nature of quantum theory, including the wave-particle duality. On the other hand, rapid advancements in computing technology in recent years have provided a new tool for investigating the nature of quantum theory. In this paper, numerical simulation is used to visualize the nonlocality implied in quantum entanglement. In particular, a Gedanken experiment of a discrete approximation of a continuous quantum space is provided, similar to the case of the dual nature of particle/wave in the double-slit experiment. The results are expected to provide a different angle for a deeper understanding of the mysterious nature of quantum correlations.

In this paper, we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

We analyze correlations between two parties sharing a general mixed two-qubit state ρ_SM in order to predict measurement outcomes. It is an interesting question as to what can be predicted about the results of two complementary projective measurements on one qubit from the measurement results on the other qubit. To quantify these predictions the complementary knowledge excessesΔK(Π_M→Π_S) and ΔK(Π′_M→Π′_S) are used, where Π_S, Π′_S resp. Π_M, Π′_M represent projective measurements on the qubit S resp. M. We derive an inequality ΔK²(Π_M→Π_S) + ΔK²(Π′_M→Π′_S)≤(B_max/2)², where B_max is the maximum violation of Bell inequalities. This inequality restricts our capability to enhance the complementary predictions for any state ρ_SM and for arbitrary Π_S, Π′_S and Π_M, Π′_M. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Frank's fundamental triangular norms T_s(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for .

We present several physical applications that were generated in the framework of quantum information science. We emphasize the crucial role played, in this approach, by a group of unitary transformations, the generalized Pauli or Heisenberg–Weyl group, and by a non-classical property, called entanglement, which appears to be a basic ingredient in Quantum Information Theory. We sketch the links between entanglement and non-locality, and discuss an analogy between entanglement and (human) relationships.

The main purpose of this review is to deal with questions related to the nature of quantum information and particularly with quantum entanglement as an important component of quantum information and computing. We will not discuss here quantum computer algorithms, like the algorithm by Shor, or their advantages and disadvantages. We only cover the material that lies at the foundations of quantum information and computing and epistemological questions. We attempt to connect the famous debate between Einstein and Bohr on quantum entanglement to some of the latest work on qubits and quantum computing and to the mathematical theorems that form the basis of these discussions, particularly the theorem of Bell. We present a critical analysis of this material and hope that this will give the reader a better understanding of what can be said with certainty about the nature of quantum information.

Pseudo-telepathy is the most recent form of rejection of locality. Many of its properties have already been discovered: for instance, the minimal entanglement, as well as the minimal cardinality of the output sets, have been characterized. This paper contains two main results. First, we prove that no bipartite pseudo-telepathy game exists, in which one of the partners receives only two questions; as a corollary, we show that the minimal "input cardinality", that is, the minimal number of questions required in a bipartite pseudo-telepathy game, is 3 × 3. Second, we study the Bell-type inequality derived from the pseudo-telepathy game known as the Magic Square game: we demonstrate that it is a tight inequality for 3 inputs and 4 outputs on each side and discuss its weak resistance to noise.

This work reviews and extends recent results concerning the distribution of entanglement, as well as nonlocality (in terms of inequality violations) in tripartite qubit systems. With recourse to a Monte Carlo generation of pure and mixed states of three-qubits, we explore several features related to the distribution of entanglement (expressed in the form of different measures of multiqubit entanglement based upon bipartitions). Also, special interest is paid to maximally entangled states (such as the GHZ for three-qubits) and W states. This study also sheds some light on the interesting relation existing between some entanglement measures and perfect state discrimination in LOCC measurements relevant to cryptographic protocols. We round off the results by studying the distribution of entanglement between Alice and Bob in a modified teleportation protocol toy model over three-qubit states.

Quantum correlations resulting in violations of Bell inequalities have generated a lot of interest in quantum information science and fundamental physics. In this paper, we address some questions that become relevant in Bell-type tests involving systems with local dimension greater than 2. For CHSH-Bell tests within 2D subspaces of such high-dimensional systems, it has been suggested that experimental violation of Tsirelson's bound indicates that more than 2D entanglement was present. We explain that the overstepping of Tsirelson's bound is due to violation of fair sampling, and can in general be reproduced by a separable state, if fair sampling is violated. For a class of Bell-type inequalities generalized to d-dimensional systems, we then consider what level of violation is required to guarantee d-dimensional entanglement of the tested state, when fair sampling is satisfied. We find that this can be used as an experimentally feasible test of d-dimensional entanglement for up to quite high values of d.

We specify the local quasi hidden variable (LqHV) model reproducing the probabilistic description of allN-partite joint von Neumann measurements on an N-qudit state. Via this local probability model, we derive a new upper bound on the maximal violation by an N-qudit state of Bell inequalities of any type (either on correlation functions or on joint probabilities) for $S$ observables per site. This new upper bound not only improves for all $N,$$S$ and d the corresponding results available for general Bell inequalities in the literature but also, for the N-qubit case with two observables per site, reduces exactly to the attainable upper bound known for quantum violations of correlation $2\times \cdots \times 2$ setting Bell inequalities in a dichotomic case.

The Bell inequalities stand at the cornerstone of the developments of quantum theory on both the foundational and applied side. The discussion started as a way to test whether the quantum description of reality is complete or not, but it developed in such a way that a new research area stemmed from it, namely quantum information. Far from being and exhausted topic, in this paper, we present a constructive and geometrically intuitive description of the local polytope and its facets in a bipartite Bell scenario with two dichotomic measurements per party.

It is the purpose of the present contribution to demonstrate that the generalization of the concept of a quantum mechanical observable from the Hermitian operator of standard quantum mechanics to a positive operator-valued measure is not a peripheral issue, allegedly to be understood in terms of a trivial nonideality of practical measurement procedures, but that this generalization touches the very core of quantum mechanics, viz. complementarity and violation of the Bell inequalities.

This is a transcript of the round table discussion, moderated by G. 't Hooft, that took place at the end of the workshop “Beyond The Quantum” in the Lorentz Center of the University of Leiden, the Netherlands, 29 May - 2 June 2006.

It displays current views on foundations of quantum mechanics.

We propose a multiple-photon absorption attack on Quantum Key Distribution protocols. In this attack, the eavesdropper (Eve) is in control of the source and sends pulses correlated in polarization (but not entangled) containing several photons at frequencies for which only multiple-photon absorptions are possible in Alice's and Bob's detectors. Whenever the number of photons from one pulse are dispatched in insufficient number to trigger a multiple-photon absorption in either channel, the pulse remains undetected. We show that this simple feature is enough to reproduce the type of statistics on the detected pulses that are considered as indicating a secure quantum key distribution in entangled-based protocols, even though the source is controlled by Eve, and we discuss possible countermeasures.

In this paper, we describe our recent work on discrete quantum theory based on Galois fields. In particular, we discuss how discrete quantum theory sheds new light on the foundations of quantum theory and we review an explicit model of super-quantum correlations we have constructed in this context. We also discuss the larger questions of the origins and foundations of quantum theory, as well as the relevance of super-quantum theory for the quantum theory of gravity.

Non-locality, i.e., some sort of instantaneous interaction or correlation determination, has been identified with the theory of Quantum Mechanics in recent times. Being in direct conflict with the basic principles of Relativity Theory, it posses a challenge. Herein various critical arguments raised in the past and judged to be particularly incisive are reviewed. These include, the identification of an error in the derivation of Bell Inequalities, the observation that Bohm inadvertently selected a non-quantum venue for experimental tests of Bell Inequalities and finally, an examination of the complexities that have rendered classical simulations of these experiments unsatisfactory.