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Spin Fidelity for Three-qubit Greenberger-Horne-Zeilinger and W States Under Lorentz Transformations

Bahram Nasr Esfahani, Mohsen Aghaee

Published Jan 1, 2010• 4 citations
OpenAlex

Abstract

Constructing the reduced density matrix for a system of three massive spin- $\frac{1}{2}$ particles described by a wave packet with Gaussian momentum distribution and a spin part in the form of GHZ or W state, the fidelity for the spin part of the system is investigated from the viewpoint of moving observers in the jargon of special relativity. Using a numerical approach, it turns out that by increasing the boost speed, the spin fidelity decreases and reaches to a non-zero asymptotic value that depends on the momentum distribution and the amount of momentum entanglement.

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Constructing the reduced density matrix for a system of three massive spin- 12\frac{1}{2}21​ particles described by a wave packet with Gaussian momentum distribution and a spin part in the form of GHZ or W state, the fidelity for the spin part of the system is investigated from the viewpoint of moving observers in the jargon of special relativity. Using a numerical approach, it turns out that by increasing the boost speed, the spin fidelity decreases and reaches to a non-zero asymptotic value that depends on the momentum distribution and the amount of momentum entanglement.
Lorentz transformation of the spin density matrix as well as the momentum density matrix for a quantum system of three massive spin-1/2 particles that have an appropriate discrete momentum distribution is investigated. In a rest frame, both the spin and the momentum parts are supposed to be maximally entangled as the Greenberger–Horne–Zeilinger state. Provided the particles move with sufficiently high speeds, we can find boosts that completely remove both the spin and momentum entanglements. We can even find boosts that completely preserve both entanglements. For the considered boosts, there is no mixing between the spin and the momentum parts.
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.
In this paper, we investigated the dynamics of systems of two and three identical qubits interacting resonantly with a selected mode of a thermal field of a lossless resonator. We found solutions of the quantum time-dependent Liouville equation for various three- and two-qubit entangled states of qubits. Based on these solutions, we calculated the criterion of the qubit entanglement — fidelity. The results of numerical calculations of the fidelity showed that increasing the average number of photons in a mode leads to a decrease in the maximum degree of entanglement. It is shown that the two-qubit entangled state is more stable with respect to external noise than the three-qubit entangled Greenberger — Horne — Zeilinger states (GHZ). Moreover, a genuine entangled GHZ-state is more stable to noise than a GHZ-like entangled 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.