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Chinese Physics Letters

Tripartite Entanglement Measures of Generalized GHZ State in Uniform Acceleration*

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

Published Oct 1, 2019• 22 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/36/10/100301

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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.
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 .
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.
Using the single-mode approximation, we first calculate entanglement measures such as negativity (1–3 and 1–1 tangles) and von Neumann entropy for a tetrapartite W-Class system in noninertial frame and then analyze the whole entanglement measures, the residual π 4 and geometric Π 4 average of tangles. Notice that the difference between π 4 and Π 4 is very small or disappears with the increasing accelerated observers. The entanglement properties are compared among the different cases from one accelerated observer to four accelerated observers. The results show that there still exists entanglement for the complete system even when acceleration r tends to infinity. The degree of entanglement is disappeared for the 1–1 tangle case when the acceleration r > 0.472473. We reexamine the Unruh effect in noninertial frames. It is shown that the entanglement system in which only one qubit is accelerated is more robust than those entangled systems in which two or three or four qubits are accelerated. It is also found that the von Neumann entropy S of the total system always increases with the increasing accelerated observers, but the S κξ and S κζδ with two and three involved noninertial qubits first increases and then decreases with the acceleration parameter r , but they are equal to constants 1 and 0.811278 respectively for zero involved noninertial qubit.
According to the single-mode approximation applied to two different mo des, each associated with different uniformly accelerating reference frames, we present analytical expression of the Minkowski states for both the ground and first excited states. Applying such an approximation, we study the entanglement property of Bell and Greenberger-Horne-Zeilinger (GHZ) states formed by such states. The corresponding entanglement properties are described by studying negativity and von Neumann entropy. The degree of entanglement will be degraded when the acceleration parameters increase. We find that the greater the number of particles in the entangled system, the more stable the system that is studied by the von Neumann entropy. The present results will be reduced to those in the case of the uniformly accelerating reference frame.
We study both pentapartite GHZ and W-class states in the noninertial frame and explore their entanglement properties by carrying out the negativities including 1-4, 2-3, and 1-1 tangles, the whole entanglement measures such as algebraic and geometric averages π5 and Π5, and von Neumann entropy. We illustrate graphically the difference between the pentapartite GHZ and W-class states. We find that all 1-4, 2-3 tangles and the whole entanglements, which are observer dependent, degrade more quickly as the number of accelerated qubits increases. The entanglements of these quantities still exist even at the infinite acceleration limit. We also notice that all 1-1 tangles of pentapartite GHZ state Nαβ=NαIβ=NαIβI=0 where α,β∈(A,B,C,D,E), whereas all 1-1 tangles of the W-class state Nαβ,NαIβ and NαIβI are unequal to zero, e.g., Nαβ=0.12111 but NαIβ and NαIβI disappear at r>0.61548 and r>0.38671, respectively. We notice that the entanglement of the pentapartite GHZ and W-class quantum systems decays faster as the number of accelerated particles increases. Moreover, we also illustrate the difference of von Neumann entropy between them and find that the entropy in the pentapartite W-class state is greater than that of GHZ state. The von Neumann entropy in the pentapartite case is more unstable than those of tripartite and tetrapartite subsystems in the noninertial frame.
In this work we present the entanglement measures of a tripartite W-State entangled system in noninertial frame through the coordinate transformation between Minkowski and Rindler. Two cases are considered, i.e., when one qubit goes in a uniform acceleration a and the others remain stationary and when two qubits undergo in a uniform acceleration and while the other is stationary. The analytical negativities for one-tangle, two-tangle and π -tangle in total are not written out explicitly except for some special cases due to complicated expressions, but we illustrate them in graphics and study their dependencies on the acceleration parameters r b and r c . We find that the negativities of the one-tangle, two-tangle and π -tangle decrease with the acceleration parameters except for the constant N A B = N B A . The negativity N C I ( A B ) and the π C I decrease faster than N A ( B C I ) and π A ( π B ) . The negativities N A C I ( N B ( A C I ) )and N B I C I decrease faster than those N A B and N A B I , respectively. It is interesting to see there exist turning points for the negativity N A ( B I C I ) at the coordinate position ( r b , r c ) with r b = r c , which implies that the N A ( B I C I ) has minimum value when Bob and Charlie are on the same position and the degree of the entanglement of the subsystem ρ A ( B I C I ) becomes smallest. The von Neumann entropy of tripartite system densities ρ A B C I and ρ A B I C I are obtained analytically. We notice that they all increase with the acceleration parameters r b and r c .