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arXiv:1201.0492 · Physical Review A

Relativistic quantum nonlocality for the three-qubit Greenberger-Horne-Zeilinger state

Shahpoor Moradi

Published Feb 27, 2008• 17 citations
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Abstract

Lorentz transformation of the three-qubit Greenberger-Horne-Zeilinger (GHZ) state is studied. Also we obtain the relativistic spin joint measurement for the transformed state. Using these results it is shown that Bell's inequality is maximally violated for the three-qubit GHZ state in the relativistic regime. For ultrarelativistic particles we obtain the critical value for boost speed, which Bell's inequality is not violated for velocities smaller than this value. We also show that in the ultrarelativistic limit Bell's inequality is maximally violated for the GHZ state.

DOI
10.1103/physreva.77.024101

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Lorentz transformation of the three-qubit Greenberger-Horne-Zeilinger (GHZ) state is studied. Also we obtain the relativistic spin joint measurement for the transformed state. Using these results it is shown that Bell's inequality is maximally violated for the three-qubit GHZ state in the relativistic regime. For ultrarelativistic particles we obtain the critical value for boost speed, which Bell's inequality is not violated for velocities smaller than this value. We also show that in the ultrarelativistic limit Bell's inequality is maximally violated for the GHZ state.
Bell-like inequalities have been used in order to distinguish non-local quantum pure states by various authors. The behavior of such inequalities under Lorentz transformation (LT) has been a source of debate and controversies in the past. In this paper, we consider the two most commonly studied three-particle pure states, that of W and Greenberger–Horne–Zeilinger (GHZ) states which exhibit distinctly different types of entanglement. We discuss the various types of three-particle inequalities used in previous studies and point to their corresponding shortcomings and strengths. Our main result is that if one uses Czachor’s relativistic spin operator and Svetlichny’s inequality as the main measure of non-locality and uses the same angles in the rest frame (S) as well as the moving frame ([Formula: see text]), then maximally violated inequality in S will decrease in the moving frame, and will eventually lead to lack of non-locality (i.e. satisfaction of inequality) in the [Formula: see text] limit. This is shown for both the GHZ and W states and in two different configurations which are commonly studied (Cases 1 and 2). Our results are in line with a more familiar case of two particle case. We also show that the satisfaction of Svetlichny’s inequality in the [Formula: see text] limit is independent of initial particles’ velocity. Our study shows that whenever we use Czachor’s relativistic spin operator, results draws a clear picture of three-particle non-locality making its general properties consistent with previous studies on two-particle systems regardless of the W state or the GHZ state is involved. Throughout the paper, we also address the results of using Pauli’s operator in investigating the behavior of [Formula: see text] under LT for both of the GHZ and W states and two cases (Cases 1 and 2). Our investigation shows that the violation of [Formula: see text] in moving frame depends on the particle’s energy in the lab frame, which is in agreement with some previous works on two and three-particle systems. Our work may also help us to classify the results of using Czachor’s and Pauli’s operators to describe the spin entanglement and thus the system spin in relativistic information theory.
Entanglement of tripartite spin states under Lorentz transformations is studied in the context of Bell’s inequality and positive partial transpose criterion. First the relativistic analogue of Bell’s inequality is discussed for three qubit states by explicit calculation of the Wigner rotation. We use the relativistic invariant spin operator which is related to the Pauli-Lubanski pseudo vector. For observers at rest the Bell’s inequality is speed-independent and maximally violated. For moving observers it’s shown that Bell’s inequality is violated and the amount of violation depends on the boost speed. We show that in ultrarelativistic limit Bell’s inequality is still maximally violated. We also obtained the critical value for satisfying Bell’s inequality. The critical value of boost speed for violation of inequality for particles moving in the center of mass frame is greater than that for particles moving with the same momentum. Second we investigate the entanglement distillability of tripartite mixed spin states under Lorentz transformations in the context of Werner states. We show that there are states that will change from distillable (entangled) into separable for a certain value of rapidity.
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.
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.
Ardehali (Phys. Rev. A, 46, 5375 (1992). 10.1103/PhysRevA.46.5375 ) derived a Bell-type inequality for an n-particle system in a Greenberger–Horne–Zeilinger (GHZ) state. This inequality helps relax the detection efficiencies required for closing the detection loophole in Bell tests. Coffman et al. (Phys. Rev. A, 61, 052306 (2000). doi: 10.1103/PhysRevA.61.052306 ) gave a simplified form of tripartite entanglement of three-qubit pure states. We use theory and experiment to investigate both results using three-qubit GHZ states with different intensities of bit-flip noise. The purpose of our work is to relate the Ardehali inequality and the tripartite concurrence of the tripartite GHZ states in a noisy environment. Our results show that the experimental values are quite consistent with theoretical predictions for three-qubit GHZ states in a bit-flip noisy environment.