Characterization of nonclassicality or quantumness of a state is fundamental to foundations of quantum mechanics and quantum information. At the heart of the problem is the question whether there exist classical systems---howsoever complicated---that can mimic a given quantum state. Whilst this has been traditionally addressed through the violation of Bell inequality or nonseparability, we show that it is possible to go beyond them, by introducing the concept of classical simulation. Focusing on the two-qubit case, we show that, while for pure states, classical simulability is equivalent to existence of a local hidden variable (LHV) model, the conditions for simulability can be weaker for mixed states, demanding what we call only a generalized LHV description. Consequently, quantum states which defy a classical simulation---which we call exceptional---may require conditions which are more stringent than violation of Bell inequalities. We illustrate these features with a number of representative examples and discuss the underlying reasons, by employing fairly simple arguments.
Characterization of nonclassicality or quantumness of a state is fundamental to foundations of quantum mechanics and quantum information. At the heart of the problem is the question whether there exist classical systems---howsoever complicated---that can mimic a given quantum state. Whilst this has been traditionally addressed through the violation of Bell inequality or nonseparability, we show that it is possible to go beyond them, by introducing the concept of classical simulation. Focusing on the two-qubit case, we show that, while for pure states, classical simulability is equivalent to existence of a local hidden variable (LHV) model, the conditions for simulability can be weaker for mixed states, demanding what we call only a generalized LHV description. Consequently, quantum states which defy a classical simulation---which we call exceptional---may require conditions which are more stringent than violation of Bell inequalities. We illustrate these features with a number of representative examples and discuss the underlying reasons, by employing fairly simple arguments.
While entanglement and violation of Bell inequalities were initially thought to be equivalent quantum phenomena, we now have different examples of entangled states whose correlations can be described by local hidden--variable models and, therefore, do not violate any Bell inequality. We provide an up to date overview of the existing local hidden--variable models for entangled quantum states, both in the bipartite and multipartite case, and discuss some of the most relevant open questions in this context. Our review covers twenty five years of this line of research since the seminal work by Werner [R. F. Werner, Phys. Rev. A 40, 8 (1989)] providing the first example of an entangled state with a local model, which in turn appeared twenty five years after the seminal work by Bell [J. S. Bell, Physics 1, 195 (1964)], about the impossibility of recovering the predictions of quantum mechanics using a local hidden--variables theory.
It is one of the most remarkable features of quantum physics that measurements on spatially separated systems cannot always be described by a locally causal theory. In such a theory, the outcomes of local measurements are determined in advance solely by some unknown (or hidden) variables and the choice of local measurements. Correlations that are allowed within the framework of a locally causal theory are termed classical. Typically, the fact that quantum mechanics does not always result in classical correlations is revealed by the violation of Bell inequalities, which are constraints that have to be satisfied by any classical correlations. It has been known for a long time that entanglement is necessary to demonstrate nonclassical correlations, and hence a Bell inequality violation. However, since some entangled quantum states are known to admit explicit locally causal models, the exact role of entanglement in Bell inequality violation has remained obscure. This thesis provides both a comprehensive review on these issues as well as a report on new discoveries made to clarify the relationship between entanglement and Bell inequality violation. In particular, within the framework of a standard Bell experiment, i.e., a Bell inequality test that is directly performed on a single copy of a quantum state r, we have derived two algorithms to determine, respectively, a lower bound and an upper bound on the strength of correlations that r can offer for any given Bell inequality. Both of these algorithms make use convex optimization techniques in the form of a semidefinite program. By examples, we show that these algorithms can often be used in tandem, in conjunction with convexity arguments, to determine if a quantum state can offer nonclassical correlations and hence violates a given Bell inequality. On the other hand, since a standard Bell experiment typically involves measurements over many copies of the quantum systems, we have also investigated the possibility of enhancing the strength of nonclassical correlation by, instead, performing collective measurements on multiple copies of the quantum systems. Our findings show that even without postselection, such joint measurements may also lead to stronger nonclassical correlations, and hence a better Bell inequality violation. Meanwhile, previous studies have indicated that entangled state admitting locally causal models may still lead to observable nonclassical correlations if, prior to a standard Bell experiment, the state is subjected to some appropriate local preprocessing. This phenomenon of hidden nonlocality was discovered more than a decade ago, but to date, it is still not known if all entangled states can demonstrate nonclassical correlations through these more sophisticated Bell experiments. A key result in this thesis then consists of showing that for all bipartite entangled states, observable nonclassical correlations, in the form of a Bell-CHSH inequality violation, can indeed be derived if we allow both local preprocessing and the usage of shared ancillary state which by itself does not violate the Bell-CHSH inequality. This establishes a kind of equivalence between bipartite entanglement and states that cannot be simulated by classical correlations. In summary, for a standard Bell experiment where no local preprocessing on a quantum state r is allowed, we have provided two algorithms that can be used in tandem to determine if r can be simulated by a locally causal theory, whereas in the scenario where local preprocessing is allowed, we have demonstrated that bipartite entangled states are precisely those which cannot always be simulated classically.
We consider Bell tests in which the distant observers can perform local filtering before testing a Bell inequality. Notably, in this setup, certain entangled states admitting a local hidden variable model in the standard Bell scenario can nevertheless violate a Bell inequality after filtering, displaying so-called hidden nonlocality. Here we ask whether all entangled states can violate a Bell inequality after well-chosen local filtering. We answer this question in the negative by showing that there exist entangled states without hidden nonlocality. Specifically, we prove that some two-qubit Werner states still admit a local hidden variable model after any possible local filtering on a single copy of the state.
We consider Bell tests in which the distant observers can perform local filtering before testing a Bell inequality. Notably, in this setup, certain entangled states admitting a local hidden variable model in the standard Bell scenario can nevertheless violate a Bell inequality after filtering, displaying so-called hidden nonlocality. Here we ask whether all entangled states can violate a Bell inequality after well-chosen local filtering. We answer this question in the negative by showing that there exist entangled states without hidden nonlocality. Specifically, we prove that some two-qubit Werner states still admit a local hidden variable model after any possible local filtering on a single copy of the state.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
This paper demonstrates the capability to simulate EPR experiments in a realistic way (i.e.without transmission of information between the source and the detector) and confirms even more the Bell's theorem, by demonstrating that effectively non-local variables are required to violate the Bell's inequality.Unlike all EPR experiments, this modeling instead of simply noting the inadequacy of a local modeling proposes a new way of non-local idealization.These simulations work on spreadsheet by idealizing objects with a non-linear notion, the extended object.This allows to set up a new principle of locality (or non-locality) and a new meaning on the loopholes of EPR experiments.It sheds new light on several concepts of Quantum Mechanics (QM): the quantum states (would represent an equivalence class of many different indistinguishable objects seen as equivalent in terms of measurement), the wave function (would represent a probability on this equivalence class seen as a single set despite different probabilities of happening for the individual objects composing this equivalence class), the measurement (in our simulation, the measured object is not modified), the superposition of states (in our simulation, Schrödinger's cat is never both dead and alive).These simulations lead to several kinds of EPR experiments with violations of the Bell's inequalities with values different from the QM, allow to realize entanglements with more than 2 objects (whatever the number of Alice and Bob) or even with more than 2 results (beyond the only "+1" or "-1" outputs).Many questions remain to be explored in the physics domain on quantum phenomena (teleportation, encryption, quantum computing, etc.), but this theoretical approach also reveals the necessity of developing new non-linear tools in mathematics domain and seems to show that QM could finally be a limit case of a more fundamental nonlinear physics theory founded on these How to cite this paper: Le Corre, S.