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arXiv:1407.5958

Local models and hidden nonlocality in Quantum Theory

Leonardo Guerini

Published Jul 22, 2014• 0 citations
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Abstract

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 $d\geq5$, 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.

DOI
10.48550/arxiv.1407.5958

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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≥5d\geq5d≥5, the Gisin-Degorre's local model for a Werner state of dimension d=2d=2d=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.
The nonlocality of certain quantum states can be revealed by using local filters before performing a standard Bell test. This phenomenon, known as hidden nonlocality, has been so far demonstrated only for a restricted class of measurements, namely, projective measurements. Here, we prove the existence of genuine hidden nonlocality. Specifically, we present a class of two-qubit entangled states, for which we construct a local model for the most general local measurements, and show that the states violate a Bell inequality after local filtering. Hence, there exist entangled states, the nonlocality of which can be revealed only by using a sequence of measurements. Finally, we show that genuine hidden nonlocality can be maximal. There exist entangled states for which a sequence of measurements can lead to maximal violation of a Bell inequality, while the statistics of nonsequential measurements is always local.
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.
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.
The violation of the Bell inequalities implies that quantum mechanics cannot be interpreted using any local hidden variable theory. However, particular quantum states that can originally be described using a local hidden variable model surprisingly exhibit nonlocality by employing local filters before a standard Bell test is performed. This is referred to as hidden nonlocality. In this study, we provide the experimental demonstration of hidden nonlocality through linear optics towards the local states which are put forword by Hirsch et al., PRL 111, 160402 (2013). A class of local states is generated through a spontaneous parametric down-conversion process, and the violation of the Clauser-Horne-Shimony-Holt (CHSH)-Bell inequality is observed by applying local filters. Our experimental results confirm the superiority of local filters, and throw light on deep understanding the intriguing phenomena of hidden nonlocality and local states.