Using the single-mode approximation, we study entanglement measures including two independent quantities; i.e., negativity and von Neumann entropy for a tripartite generalized Greenberger–Horne–Zeilinger (GHZ) state in noninertial frames. Based on the calculated negativity, we study the whole entanglement measures named as the algebraic average π 3 -tangle and geometric average Π 3 -tangle. We find that the difference between them is very small or disappears with the increase of the number of accelerated qubits. The entanglement properties are discussed from one accelerated observer and others remaining stationary to all three accelerated observers. The results show that there will always exist entanglement, even if acceleration r arrives to infinity. The degree of entanglement for all 1–1 tangles are always equal to zero, but 1–2 tangles always decrease with the acceleration parameter r. We notice that the von Neumann entropy increases with the number of the accelerated observers and S κ I ζ I (κ, ζ ∈ (A, B, C)) first increases and then decreases with the acceleration parameter r. This implies that the subsystem ρ κ I ζ I is first more disorder and then the disorder will be reduced as the acceleration parameter r increases. Moreover, it is found that the von Neumann entropies S ABCI, S ABICI and S AIBICI always decrease with the controllable angle θ, while the entropies of the bipartite subsystems S 2−2non (two accelerated qubits), S 2-1non (one accelerated qubit) and S 2-0non (without accelerated qubit) first increase with the angle θ and then decrease with it.

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Using the single-mode approximation we study the quantum correlations of a free Dirac field on the Werner state in the noninertial frame. We present analytical quantum discord D and its geometric measurement DG. We notice that the quantum discord still exists for the system ρARI even in the acceleration limit. We also calculate the entanglement of formation (EOF) and the geometric measure of entanglement (EG) to compare the behaviors of the entanglement with the quantum discord. The entanglement is nonzero for the fidelity of the Werner state with a Bell state F > 0.5, while the quantum discord is equal to zero only for F = 0.25. We also observe that the quantum discord is more robust to the change of acceleration than the entanglement, which is more sensitive to the acceleration.

The Schrödinger equation ψ ′′ (x) + κ 2 ψ(x) = 0 where κ 2 = k 2 − V (x) is rewritten as a more popular form of a second order differential equation through taking a similarity transformation ψ(z) = φ(z)u(z) with z = z(x). The Schrödinger invariant I S (x) can be calculated directly by the Schwarzian derivative {z, x} and the invariant I(z) of the differential equation u zz + f (z)u z + g(z)u = 0. We find an important relation for moving particle as ∇ 2 = −I S (x) and thus explain the reason why the Schrödinger invariant I S (x) keeps constant. As an illustration, we take the typical Heun differential equation as an object to construct a class of soluble potentials and generalize the previous results through choosing different ρ = z ′ (x) as before. We get a more general solution z(x) * E-mail address: dongsh2@yahoo. com through integrating (z ′ ) 2 = α 1 z 2 + β 1 z + γ 1 directly and it includes all possibilities for those parameters. Some particular cases are discussed in detail.