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Tetrapartite entanglement measures of GHZ state with nonuniform acceleration

Qian Dong, Guo-Hua Sun, Mohamad M. Toutounji, Shi‐Hai Dong

Published Oct 14, 2019• 14 citations
Publisher
DOI
10.1016/j.ijleo.2019.163487

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Using a single-mode approximation, we carry out the entanglement measures, e.g., the negativity and von Neumann entropy when a tetrapartite generalized GHZ state is treated in a noninertial frame, but only uniform acceleration is considered for simplicity. In terms of explicit negativity calculated, we notice that the difference between the algebraic average π 4 and geometric average Π 4 is very small with the increasing accelerated observers and they are totally equal when all four qubits are accelerated simultaneously. The entanglement properties are discussed from one accelerated observer to all four accelerated observers. It is shown that the entanglement still exists even if the acceleration parameter r goes to infinity. It is interesting to discover that all 1-1 tangles are equal to zero, but 1-3 and 2-2 tangles always decrease when the acceleration parameter r increases. We also study the von Neumann entropy and find that it increases with the number of the accelerated observers. In addition, we find that the von Neumann entropy S ABCDI, S ABCIDI, S ABICIDI and S AIBICIDI always decrease with the controllable angle θ, while the entropies S 3 – 3 non, S 3 – 2 non, S 3 – 1 non and S 3 – 0 non first increase with the angle θ and then decrease with it.

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Tetrapartite entanglement measures of GHZ state with uniform acceleration

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Abstract Using the single-mode approximation, we calculate entanglement measures such as negativity (1 − 3, 1 − 1 and 2 − 2 tangles) and von Neumann entropy for a tetrapartite Greenberger–Horne–Zeilinger system in a non-inertial frame. We then analyze the whole entanglement measures, the residual π 4 and geometric Π 4 average of tangles. We find that the difference between them is very small or disappears with the increasing accelerated observers. The entanglement properties are compared among the different cases from one accelerated observer and others remaining stationary to all four accelerated observers. The results presented here show that entanglement still exists for the complete system even when acceleration r tends to infinity. The degree of entanglement is always equal to zero for the 1 − 1 tangle case. We also study the negativity 2 − 2 tangle for completeness. We reexamine the existence of the Unruh effect in non-inertial frames. It is also found that the von Neumann entropy increases with the increasing accelerated observers, and S κ I ζ I and S κ I ζ I δ I first increase and then decrease with the acceleration parameter r , e.g. S κ ζ ( max ) = 1.65853 at r = 0.599881 and S κ ζ δ ( max ) = 2.28231 at r = 0.676196. This implies that the subsystems ρ κ I ζ I and ρ κ I ζ I δ I are at first more disordered and then the disorder is reduced with the increasing acceleration parameter r .

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We present the entanglement measures of a tetrapartite W-class entangled system in a noninertial frame, where the transformation between Minkowski and Rindler coordinates is applied. Two cases are considered. First, when one qubit has uniform acceleration whilst the other three remain stationary. Second, when two qubits have nonuniform accelerations and the others stay inertial. The 1–1 tangle, 1–3 tangle, and whole entanglement measurements π 4 and Π 4 , are studied and illustrated with graphics through their dependence on the acceleration parameter r d for the first case and r c and r d for the second case. It is found that the negativities (1–1 tangle and 1–3 tangle) and π-tangle decrease when the acceleration parameter r d or in the second case r c and r d increase, remaining a nonzero entanglement in the majority of the results. This means that the system will be always entangled except for special cases. It is shown that only the 1–1 tangle for the first case vanishes at infinite accelerations, but for the second case the 1–1 tangle disappears completely when r > 0.472473 . An analytical expression for the von Neumann information entropy of the system is found and we note that it increases with the acceleration parameter.

Chinese Physics B•Jul 1, 2019•17 citations
Using a single-mode approximation, we carry out the entanglement measures, e.g., the negativity and von Neumann entropy when a tetrapartite generalized GHZ state is treated in a noninertial frame, but only uniform acceleration is considered for simplicity. In terms of explicit negativity calculated, we notice that the difference between the algebraic average π 4 and geometric average Π 4 is very small with the increasing accelerated observers and they are totally equal when all four qubits are accelerated simultaneously. The entanglement properties are discussed from one accelerated observer to all four accelerated observers. It is shown that the entanglement still exists even if the acceleration parameter r goes to infinity. It is interesting to discover that all 1-1 tangles are equal to zero, but 1-3 and 2-2 tangles always decrease when the acceleration parameter r increases. We also study the von Neumann entropy and find that it increases with the number of the accelerated observers. In addition, we find that the von Neumann entropy S ABCDI, S ABCIDI, S ABICIDI and S AIBICIDI always decrease with the controllable angle θ, while the entropies S 3 – 3 non, S 3 – 2 non, S 3 – 1 non and S 3 – 0 non first increase with the angle θ and then decrease with it.
Abstract Using the single-mode approximation, we calculate entanglement measures such as negativity (1 − 3, 1 − 1 and 2 − 2 tangles) and von Neumann entropy for a tetrapartite Greenberger–Horne–Zeilinger system in a non-inertial frame. We then analyze the whole entanglement measures, the residual π 4 and geometric Π 4 average of tangles. We find that the difference between them is very small or disappears with the increasing accelerated observers. The entanglement properties are compared among the different cases from one accelerated observer and others remaining stationary to all four accelerated observers. The results presented here show that entanglement still exists for the complete system even when acceleration r tends to infinity. The degree of entanglement is always equal to zero for the 1 − 1 tangle case. We also study the negativity 2 − 2 tangle for completeness. We reexamine the existence of the Unruh effect in non-inertial frames. It is also found that the von Neumann entropy increases with the increasing accelerated observers, and S κ I ζ I and S κ I ζ I δ I first increase and then decrease with the acceleration parameter r , e.g. S κ ζ ( max ) = 1.65853 at r = 0.599881 and S κ ζ δ ( max ) = 2.28231 at r = 0.676196. This implies that the subsystems ρ κ I ζ I and ρ κ I ζ I δ I are at first more disordered and then the disorder is reduced with the increasing acceleration parameter r .
We present the entanglement measures of a tetrapartite W-class entangled system in a noninertial frame, where the transformation between Minkowski and Rindler coordinates is applied. Two cases are considered. First, when one qubit has uniform acceleration whilst the other three remain stationary. Second, when two qubits have nonuniform accelerations and the others stay inertial. The 1–1 tangle, 1–3 tangle, and whole entanglement measurements π 4 and Π 4 , are studied and illustrated with graphics through their dependence on the acceleration parameter r d for the first case and r c and r d for the second case. It is found that the negativities (1–1 tangle and 1–3 tangle) and π-tangle decrease when the acceleration parameter r d or in the second case r c and r d increase, remaining a nonzero entanglement in the majority of the results. This means that the system will be always entangled except for special cases. It is shown that only the 1–1 tangle for the first case vanishes at infinite accelerations, but for the second case the 1–1 tangle disappears completely when r > 0.472473 . An analytical expression for the von Neumann information entropy of the system is found and we note that it increases with the acceleration parameter.