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New Journal of Physics

Exploring nonlocal correlations in arbitrarily high-dimensional systems

Artur Barasiński

Published Nov 1, 2024• 2 citations
Publisher

Abstract

Abstract The utilization of higher-dimensional systems holds promise for unveiling novel phenomena and enhancing the efficacy of practical tasks, such as entanglement-based quantum cryptography. However, detecting quantum correlations within qudit systems poses a formidable challenge. In this study, we delve into the intricacies of Bell’s nonlocality within higher-dimensional systems. We introduce a novel correlation function and leverage it to construct a family of Bell inequalities adapted for quantum systems of arbitrary dimensionality. By gauging the probability of local realism violation under random measurements as our primary metric, we explore the relation between pure state entanglement and nonlocality. Our findings align closely with predictions derived from the comprehensive polytope analysis via linear programming methods. While our focus predominantly centers on the two-qudit scenario, our methodology readily extends to the N-partite case, accommodating an arbitrary number of measurements per party.

DOI
10.1088/1367-2630/ad9170

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Abstract The utilization of higher-dimensional systems holds promise for unveiling novel phenomena and enhancing the efficacy of practical tasks, such as entanglement-based quantum cryptography. However, detecting quantum correlations within qudit systems poses a formidable challenge. In this study, we delve into the intricacies of Bell’s nonlocality within higher-dimensional systems. We introduce a novel correlation function and leverage it to construct a family of Bell inequalities adapted for quantum systems of arbitrary dimensionality. By gauging the probability of local realism violation under random measurements as our primary metric, we explore the relation between pure state entanglement and nonlocality. Our findings align closely with predictions derived from the comprehensive polytope analysis via linear programming methods. While our focus predominantly centers on the two-qudit scenario, our methodology readily extends to the N-partite case, accommodating an arbitrary number of measurements per party.
The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from d=2d=2d=2 up to d=10d=10d=10. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension ddd grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviors. Furthermore, in the latter approach the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison to maximally entangled states, so no anomaly is observed. Finally, the nonlocality of states with nonmaximal Schmidt rank is investigated.
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We present a detailed investigation of minimum detection efficiencies, below which local realism cannot be violated by any quantum system of any dimension in bipartite Bell experiments. Lower bounds on these minimum detection efficiencies are determined with the help of linear programming techniques. Our approach is based on the observation that any possible bipartite quantum correlation originating from a quantum state in an arbitrary dimensional Hilbert space is sandwiched between two probability polytopes, namely the local (Bell) polytope and a corresponding nonlocal no-signaling polytope. Numerical results are presented demonstrating the dependence of these lower bounds on the numbers of inputs and outputs of the bipartite physical system.
Abstract This paper investigates the problem of nonlocality detection for high-dimensional quantum states. To overcome the complexity of measurement settings and Bell-operator optimization in high-dimensional systems, we propose a nonlocality detection framework based on dimensional reduction and chained Bell inequalities. By employing local projection operators, arbitrary high-dimensional bipartite quantum states are mapped onto effective two-qubit subsystems. We prove that if any projected subsystem violates a chained Bell inequality, then the original high-dimensional quantum state must be nonlocal. Furthermore, the framework is generalized to multipartite quantum systems. By projecting multipartite high-dimensional states onto effective multiqubit states and applying Svetlichny inequalities, sufficient criteria for detecting genuine multipartite nonlocality are established. Our approach significantly reduces the complexity of nonlocality detection in high-dimensional systems and provides a scalable theoretical framework for studying nonlocal correlations in complex quantum systems.
The aim of this thesis is to investigate quantum entanglement and quantum nonlocality of bipartite finite-dimensional systems (bipartite qudits). Entanglement is one of the most fascinating non-classical features of quantum theory, and besides its impact on our view of the world, it can be exploited for applications such as quantum cryptography and quantum computing. (...) Although entanglement and nonlocality are ordinarily regarded as one and the same, under close consideration this cannot be taken for granted. The reason for this is that entanglement is defined by the mathematical structure of a quantum state in a composite Hilbert space, whereas nonlocality signifies that the statistical behaviour of a system cannot be described by a local realistic theory. For the latter it is essential that the correlation probabilities of such theories obey so-called Bell inequalities, which are violated for certain quantum states. The main focus of this thesis is on the comparison of both properties with the objective of understanding their relation. (...) Because of the fact that the correlation probabilities in general depend on the measurement settings it is necessary to optimise these in order to reveal nonlocality. This problem is solved for a particular Bell inequality (CGMLP) by means of a self-developed numerical search algorithm. The method is then applied to density matrices of a subspace spanned by the projectors of maximally entangled two-qudit states. This set of states (...) serves to visualise and analyse the state space geometrically.
Abstract For the maximal violation of all Bell inequalities by an arbitrary pure two-qudit state of any dimension, we derive a new lower bound expressed via the concurrence of this pure state. This new lower bound and the upper bound on the maximal Bell violation, found in Loubenets and Namkung (2022 J. Phys. A: Math. Theor. 55 285301) and also expressed via the concurrence, analytically quantify Bell nonlocality of a pure two-qudit state via its entanglement, in particular, prove explicitly that entanglement of a pure two-qudit state is necessary and sufficient for its Bell nonlocality. By re-visiting the pure two-qubit case, we also find and rigorously prove the new results on correlation properties of an arbitrary pure two-qubit state.