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EPR Paradox, Nonlocality, and Entanglement in Multi-qubit Systems

Raja Emlik

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

Bell inequalities were formulated by John Bell to test the possible violation of local realistic theories by quantum mechanical systems. It was shown that entangled quantu-m states of multiple particles violate various Bell’s inequalities. This proved that quan-tum mechanics allows correlations between spatially separated systems that have no classical analogue. The main focus of this work is to investigate genuine multiqubit non-locality in families of entangled 3 and 4-qubit pure states by studying a Bell-type inequality that is violated only if all qubits are non-locally correlated. We numerically study the relationship between entanglement and violation of the Svetlichny Bell-type inequality. We analyze non-local correlations in 3-qubit generalized Greenberger-Hor-ne-Zeilinger (GHZ) states, maximal slice (MS) states, and W states. Our studies show that the correlations exhibited by three particles cannot in general be described by hid-den variable theories with at most two-particle non-locality. However, some 3-qubit entangled states do not violate the Svetlichny’s inequality. We then extend our analysis to 4-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states, maximal slice (MS) states, and W states. The results are similar to the 3-qubit case for GHZ and MS states. The range of parameters for which we see a violation is the same for the 3 and 4-qubit GHZ states. However, the 4-qubit W states do not violate Bell-type inequality, unlike the 3-qubit W states. Our results show the complex nature of multiqubit entang-lement and non-locality and provide tools for designing useful quantum communica-tion tasks.

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Bell inequalities were formulated by John Bell to test the possible violation of local realistic theories by quantum mechanical systems. It was shown that entangled quantu-m states of multiple particles violate various Bell’s inequalities. This proved that quan-tum mechanics allows correlations between spatially separated systems that have no classical analogue. The main focus of this work is to investigate genuine multiqubit non-locality in families of entangled 3 and 4-qubit pure states by studying a Bell-type inequality that is violated only if all qubits are non-locally correlated. We numerically study the relationship between entanglement and violation of the Svetlichny Bell-type inequality. We analyze non-local correlations in 3-qubit generalized Greenberger-Hor-ne-Zeilinger (GHZ) states, maximal slice (MS) states, and W states. Our studies show that the correlations exhibited by three particles cannot in general be described by hid-den variable theories with at most two-particle non-locality. However, some 3-qubit entangled states do not violate the Svetlichny’s inequality. We then extend our analysis to 4-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states, maximal slice (MS) states, and W states. The results are similar to the 3-qubit case for GHZ and MS states. The range of parameters for which we see a violation is the same for the 3 and 4-qubit GHZ states. However, the 4-qubit W states do not violate Bell-type inequality, unlike the 3-qubit W states. Our results show the complex nature of multiqubit entang-lement and non-locality and provide tools for designing useful quantum communica-tion tasks.
We investigate genuine multiqubit nonlocality in families of entangled 3- and 4-qubit pure states by analyzing a Bell-type inequality that is violated only if all qubits are nonlocally correlated. We present detailed numerical studies of the relationship between entanglement and violation of the Svetlichny Bell-type inequality in an experimentally accessible set of 3-qubit pure states, and identify the special nonlocality property of the maximal slice states in the space of all 3-qubit pure states. We also analyze nonlocal correlations in 3-qubit generalized Greenberger–Horne–Zeilinger (GHZ) states and extend our analysis to the case of 4-qubit generalized GHZ states. We show that like the 3-qubit case, some 4-qubit generalized GHZ states do not violate a Bell inequality that tests for genuine 4-qubit nonlocality. Furthermore, the location of the boundary between the states that do violate the inequality and those that do not is the same for the 3- and 4-qubit generalized GHZ states.
Entangled states of multiple qubits can violate Bell-type inequalities indicating nonlocal behavior of multiqubit quantum correlations. We analyze the relation between multipartite entanglement and genuine multipartite nonlocality, characterized by Svetlichny inequality violations, for two families of N − qubit states. We show that for the generalized Greenberger-Horne-Zeilinger family of states, Svetlichny inequality is not violated when the n − tangle is less than 1 / 2 for any even number of qubits. On the other hand, the maximal slice states always violate the Svetlichny inequality when n − tangle is nonzero, and the violation increases monotonically with tangle. Our work generalizes the relations between tangle and Svetlichny inequality violations previously derived for three qubits.
We analyze the relationship between tripartite entanglement and genuine tripartite nonlocality for three-qubit pure states in the Greenberger-Horne-Zeilinger class. We consider a family of states known as the generalized Greenberger-Horne-Zeilinger states and derive an analytical expression relating the three-tangle, which quantifies tripartite entanglement, to the Svetlichny inequality, which is a Bell-type inequality that is violated only when all three qubits are nonlocally correlated. We show that states with three-tangle less than 1/2 do not violate the Svetlichny inequality. On the other hand, a set of states known as the maximal slice states does violate the Svetlichny inequality, and exactly analogous to the two-qubit case, the amount of violation is directly related to the degree of tripartite entanglement. We discuss further interesting properties of the generalized Greenberger-Horne-Zeilinger and maximal slice states.
Entanglement, the Einstein-Podolsky-Rosen (EPR) paradox and Bell’s failure of local-hiddenvariable (LHV) theories are three historically famous forms of “quantum nonlocality”. We give experimental criteria for these three forms of nonlocality in multi-particle systems, with the aim of better understanding the transition from microscopic to macroscopic nonlocality. We examine the nonlocality of N separated spin J systems. First, we obtain multipartite Bell inequalities that address the correlation between spin values measured at each site, and then we review spin squeezing inequalities that address the degree of reduction in the variance of collective spins. The latter have been particularly useful as a tool for investigating entanglement in Bose-Einstein condensates (BEC). We present solutions for two topical quantum states: multi-qubit Greenberger-Horne-Zeilinger (GHZ) states, and the ground state of a two-well BEC.
In this paper, we theoretically study the relation between concurrence and nonlocality depicted by Bell-type inequality violation of quantum mechanics prediction versus local realism prediction for the GHZ (Greenberger -Horne-Zeilinger) class states. Analytical expressions of concurrence, violations of the Mermin inequality and the Svetlichny inequality are obtained. Through numerical calculations, the relationship between entanglement and nonlocality of GHZ-class states is discussed. Our results show that the concurrence is consistent with the degree of nonlocality described by violations of the two Bell-type inequalities of GHZ-class states. The Bell operator and its parameters can obviously reveal the nonlocal features of quantum states.
Quantum physics violates the basic assumptions of local realism. Particles can be entangled with each other in ways that allow correlations stronger than possible in classical physics. This was first realized by Einstein, Podolsky and Rosen (EPR) in 1935 and led them to conclude that quantum theory was incomplete. It took about 30 years before J. S. Bell found observable consequences of this phenomenon by showing how entangled states violate inequalities for two-particle correlations derived purely from the assumption that the world obeys local realism. Violations of Bell inequalities were first observed experimentally in the Aspect experiment, which will be discussed below. Such experiments are technically challenging and care is needed to avoid any loopholes in the experimental setup that would still permit the description of the experiments by a local realistic theory. Bell inequalities describe violations of local realism on average, meaning that each individual experimental outcome could occur in a local realistic theory. It is the statistics of these outcomes that violates local realism. In contrast, another class of quantum states, so-called Greenberger–Horne–Zeilinger (GHZ) states, which contain three particles, can be shown to violate local realism in each measurement outcome. We will also discuss this class of states and experiments by Zeilinger's group, which demonstrated this type of violation of local realism for the first time. Our aim here is not to discuss the exact relations between quantum entanglement, non-locality and realism, and their physical and philosophical implications. While this is a highly interesting topic which attracts a lot of attention from researchers, such an exploration would go beyond the scope of this book.