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arXiv:0802.1238 · Physical Review A

Degradation of nonmaximal entanglement of scalar and Dirac fields in noninertial frames

Qiyuan Pan, Jiliang Jing

Published Feb 8, 2008• 96 citations
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Abstract

The entanglement between two modes of free scalar and Dirac fields as seen by two relatively accelerated observers has been investigated. It is found that the same initial entanglement for an initial state parameter $\ensuremath{\alpha}$ and its ``normalized partner'' $\sqrt{1\ensuremath{-}{\ensuremath{\alpha}}^{2}}$ will be degraded by the Unruh effect along two different trajectories except for the maximally entangled state, which just shows the inequivalence of the quantization for a free field in Minkowski and Rindler coordinates. In the infinite-acceleration limit, the state does not have distillable entanglement for any $\ensuremath{\alpha}$ for the scalar field, but always remains entangled to a degree that is dependent on $\ensuremath{\alpha}$ for the Dirac field. It is also interesting to note that in this limit the mutual information equals just half of the initial mutual information; this result is independent of $\ensuremath{\alpha}$ and the type of field.

DOI
10.1103/physreva.77.024302

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The entanglement between two modes of free scalar and Dirac fields as seen by two relatively accelerated observers has been investigated. It is found that the same initial entanglement for an initial state parameter \ensuremathα\ensuremath{\alpha}\ensuremathα and its ``normalized partner'' 1\ensuremath−\ensuremathα2\sqrt{1\ensuremath{-}{\ensuremath{\alpha}}^{2}}1\ensuremath−\ensuremathα2​ will be degraded by the Unruh effect along two different trajectories except for the maximally entangled state, which just shows the inequivalence of the quantization for a free field in Minkowski and Rindler coordinates. In the infinite-acceleration limit, the state does not have distillable entanglement for any \ensuremathα\ensuremath{\alpha}\ensuremathα for the scalar field, but always remains entangled to a degree that is dependent on \ensuremathα\ensuremath{\alpha}\ensuremathα for the Dirac field. It is also interesting to note that in this limit the mutual information equals just half of the initial mutual information; this result is independent of \ensuremathα\ensuremath{\alpha}\ensuremathα and the type of field.
We analyze the entanglement between two modes of a free Dirac field as seen by two relatively accelerated parties. The entanglement is degraded by the Unruh effect and asymptotically reaches a nonvanishing minimum value in the infinite acceleration limit. This means that the state always remains entangled to a degree and can be used in quantum information tasks, such as teleportation, between parties in relative uniform acceleration. We analyze our results from the point of view afforded by the phenomenon of entanglement sharing and in terms of recent results in the area of multiqubit complementarity.
The entanglement between two modes of a free scalar field (or a free Dirac field) is investigated in noninertial frames under the condition that they are initially prepared in a Werner state. It is found that the entanglement degradation induced by the Unruh effect highly depends on the initial purity f and the type of fields. For a fixed acceleration r , the critical values of f for the states entangled or separable are obtained in both scalar and Dirac fields. Finally, we make a comparison between the entanglement and the quantum discord and show that quantum discord is more robust than entanglement in our models.
The π-tangle is used to study the behavior of entanglement of a nonmaximal tripartite state of both Dirac and scalar fields in accelerated frame. For Dirac fields, the degree of degradation with acceleration of both one-tangle of accelerated observer and π-tangle, for the same initial entanglement, is different by just interchanging the values of probability amplitudes. A fraction of both one-tangles and the π-tangle always survives for any choice of acceleration and the degree of initial entanglement. For scalar field, the one-tangle of accelerated observer depends on the choice of values of probability amplitudes and it vanishes in the range of infinite acceleration, whereas for π-tangle this is not always true. The dependence of π-tangle on probability amplitudes varies with acceleration. In the lower range of acceleration,its behavior changes by switching between the values of probability amplitudes and for larger values of acceleration this dependence on probability amplitudes vanishes. Interestingly, unlike bipartite entanglement, the degradation of π-tangle against acceleration in the case of scalar fields is slower than for Dirac fields.
The π-tangle is used to study the behavior of entanglement of a nonmaximal tripartite state of both Dirac and scalar fields in accelerated frame. For Dirac fields, the degree of degradation with acceleration of both one-tangle of accelerated observer and π-tangle, for the same initial entanglement, is different by just interchanging the values of probability amplitudes. A fraction of both one-tangles and the π-tangle always survives for any choice of acceleration and the degree of initial entanglement. For scalar field, the one-tangle of accelerated observer depends on the choice of values of probability amplitudes and it vanishes in the range of infinite acceleration, whereas for π-tangle this is not always true. The dependence of π-tangle on probability amplitudes varies with acceleration. In the lower range of acceleration, its behavior changes by switching between the values of probability amplitudes and for larger values of acceleration this dependence on probability amplitudes vanishes. Interestingly, unlike bipartite entanglement, the degradation of π-tangle against acceleration in the case of scalar fields is slower than for Dirac fields.
We obtain the necessary and sufficient separability and distillability conditions of mixtures of a maximally entangled state and the completely separable state in relativistic setting. In an inertial frame we study the entanglement under Wigner rotations induced by Lorentz transformations. We also investigate the mixed state entanglement of scalar and Dirac fields as seen by two relatively accelerated observers. For scalar field we show that in infinite acceleration limit the state has no longer distillable entanglement. For dirac field the entanglement in the infinite acceleration limit is finite. In both cases we show that there are states that will change from entangled into separable for a certain value of velocity or acceleration. We conclude that distillability is a relative concept, depending on the frame in which it is observed.
We obtain the necessary and sufficient separability and distillability conditions of mixtures of a maximally entangled state and the completely separable state in relativistic setting. In an inertial frame we study the entanglement under Wigner rotations induced by Lorentz transformations. We also investigate the mixed state entanglement of scalar and Dirac fields as seen by two relatively accelerated observers. For scalar field we show that in infinite acceleration limit the state has no longer distillable entanglement. For dirac field the entanglement in the infinite acceleration limit is finite. In both cases we show that there are states that will change from entangled into separable for a certain value of velocity or acceleration. We conclude that distillability is a relative concept, depending on the frame in which it is observed.