Nonlocality versus entanglement in quantum theory

The thesis is dedicated to the study of Bell nonlocality within quantum theory. More precisely, we investigate under which conditions a quantum experiment can give rise to nonlocal correlations. Our main results are no-go theorems: we construct an algorithm which can be used to prove that an entangled quantum state \textit{cannot} give rise to any nonlocal correlations in the standard Bell scenario; we exhibit a set of incompatible measurements which is nevertheless useless in any Bell test; we demonstrate that there exist entangled quantum states which do not display any hidden nonlocality.

It is known that quantum mechanics is \textit{nonlocal}, using a natural definition given by Bell. This astonishing phenomenon is allowed by the existence of entanglement, an inseparability of the global quantum state of distant parties. Although it is straightforward to see that separable quantum states cannot lead to nonlocality, that is, entanglement is necessary for nonlocality, the exact link between these two concepts is still not well understood.

In particular, there exist entangled states which cannot give rise to any nonlocal correlations in the standard Bell scenarios. These \textit{local entangled states} are proven so by constructing a local hidden-variable model which reproduces the correlations predicted by quantum mechanics for \textit{all} local measurements. Nevertheless, such a model is in general challenging to construct and this could only be done for a few particular families of states, while there is no tool for a general quantum state. In Chapter 3, we address this problem by giving an algorithm which constructs local hidden-variable models for entangled states. The algorithm converges in the limit, meaning that any local entangled state will be eventually detected. Together with standard existing techniques using Bell inequalities we therefore build an algorithm which finds whether a general state is local or nonlocal (in the standard Bell scenario). We show the practical implementation of our algorithm as well as its efficiency with explicit examples. In particular, in Chapter 4, we apply this method to the family of qubit Werner states and using a new computational technique we obtain the best known local hidden-variable model.

Another necessary condition for nonlocality in quantum mechanics is the \textit{incompatibility} of quantum measurements. Measurement incompatibility is, like entanglement, a non-classical feature of quantum theory, which basically prevents several measurements from being performed together on the same state. Recently showed to be equivalent to quantum steering, it was not known whether measurement incompatibility is a sufficient condition to nonlocality. Here we mean that for a set of incompatible measurements there would exist an entangled state and another set of incompatible measurements such that the resulting statistics is nonlocal. In Chapter 5, we show that this is not the case in general, by considering the strongest notion of incompatibility, non-joint measurability, and exhibiting a counter-example. That is, we exhibit a set of measurements which is not jointly-measurable while there exists a local hidden-variable model reproducing its statistics for any entangled state and sets of measurements used by other parties. \medbreak

Another necessary condition for nonlocality in quantum mechanics is the \textit{incompatibility} of quantum measurements. Measurement incompatibility is, like entanglement, a non-classical feature of quantum theory, which basically prevents several measurements from being performed together on the same state. Recently showed to be equivalent to quantum steering, it was not known whether measurement incompatibility is a sufficient condition to nonlocality. Here we mean that for a set of incompatible measurements there would exist an entangled state and another set of incompatible measurements such that the resulting statistics is nonlocal. In Chapter 5, we show that this is not the case in general, by considering the strongest notion of incompatibility, non-joint measurability, and exhibiting a counter-example. That is, we exhibit a set of measurements which is not jointly-measurable while there exists a local-hidden variable model reproducing its statistics for any entangled state and sets of measurements used by other parties.

One can go further than the standard Bell scenario and try to test the notion of locality in more general situations. A possible way to do it is by considering \textit{sequence} of measurements. In quantum mechanics, one can for instance allow the parties to perform some local filtering before a Bell test. In this scenario, it was shown that nonlocality can be \textit{activated}. Namely, some entangled states admitting local hidden-variable models can become nonlocal after suitable local filters, displaying so-called hidden nonlocality. An open question was whether entanglement is equivalent to nonlocality in this more general scenario. In Chapter 6, we answer to this question in the negative, by demonstrating that some entangled Werner states cannot exhibit any hidden nonlocality, that is, they remain local after arbitrary local filtering.

A few more related results are given in Chapter 7. We discuss the resources needed to classically simulate entanglement, that is, how many classical bits does one need in the most economical local hidden-variable models. It turns out that in all such models which were known, the local hidden-variable is picked in a continuous set, hence necessitating infinite classical communication. We show that this requirement can in fact be dispensed with. We construct several explicit models for entangled states which make use of finite resources. We finally show that all local entangled states can be simulated using finite resources only.

We then focus on the particular case of local hidden-state models, especially relevant in the context of quantum steering. We extend Werner's model for general two-qubit states and thereby derive a sufficient criterion for a two-qubit state to admit a local hidden-state model.

We finally discuss the multipartite scenario, where the structure of entanglement and nonlocality become richer. We show that the inequivalence between entanglement and nonlocality can be maximal by constructing a family of multipartite states which features the strongest form of entanglement, genuine multipartite entanglement, while being unable to produce the weakest form of nonlocality. We show that this result holds for any number of parties.