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arXiv:2306.17490 · Physical review. D/Physical review. D.

Reflected entropy and Markov gap in noninertial frames

Jaydeep Kumar Basak, Dimitrios Giataganas, Sayid Mondal, Wen-Yu Wen

Published Dec 19, 2023• 28 citations
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Abstract

We explore the reflected entropy and the Markov gap between two modes of a free fermionic field as observed by accelerating observers. This is done for both a bipartite system, which is described by the Bell state, and tripartite systems, which are represented by the Werner and Greenberger-Horne-Zeilinger states. The reflected entropy of the fermionic modes degrades monotonically as a result of the Unruh effect, eventually reaching a nonzero minimum value in the limit of infinite acceleration. Furthermore, we show that the Markov gap exhibits monotonic behavior with regard to acceleration in all three cases. In addition, we suggest a function for reflected entropy which decreases monotonically with decreasing Unruh temperature for all states. Finally, we confirm that the reflected entropy for our system does reduce under the partial tracing of the degrees of freedom for our states.

DOI
10.1103/physrevd.108.125009

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We explore the reflected entropy and the Markov gap between two modes of a free fermionic field as observed by accelerating observers. This is done for both a bipartite system, which is described by the Bell state, and tripartite systems, which are represented by the Werner and Greenberger-Horne-Zeilinger states. The reflected entropy of the fermionic modes degrades monotonically as a result of the Unruh effect, eventually reaching a nonzero minimum value in the limit of infinite acceleration. Furthermore, we show that the Markov gap exhibits monotonic behavior with regard to acceleration in all three cases. In addition, we suggest a function for reflected entropy which decreases monotonically with decreasing Unruh temperature for all states. Finally, we confirm that the reflected entropy for our system does reduce under the partial tracing of the degrees of freedom for our states.
We study the reflected entropy and the Markov gap for modes of a free bosonic field shared by inertial observers (Alice, Charlie) and a uniformly accelerated one (Bob), for a bipartite Bell state and the tripartite Werner (W) and Greenberger--Horne--Zeilinger (GHZ) states. The bosonic Bogoliubov transformation spans an infinite-dimensional Fock space with an unbounded squeezing parameter unlike the fermionic case. By identifying a conserved charge, we block-diagonalize the reduced density matrices into exact two-dimensional sectors, yielding closed or semi-analytic forms for all three states. Although bosonic entanglement is known to vanish asymptotically, the Alice--Bob reflected entropy instead saturates at a nonzero floor, retaining the surviving classical correlation, and converges to the value Alice shares with Bob's causally disconnected partner. Crucially, only the inter-wedge reflected entropy diverges, linearly in the squeezing parameter---the sharp distinction from the fermionic case, where it stays bounded---while the Markov gaps saturate. The construction transfers verbatim to a Schwarzschild black hole, where the saturation values become mass-independent constants.
Quantum coherence as the fundamental characteristic of quantum physics, provides the valuable resource for quantum computation in exceeding the power of classical algorithms. The exploration of quantum coherence in relativistic systems is of significance from both the fundamental points of view and practical applications. We investigate the quantum coherence of two free modes of scalar and Dirac fields as detected by two relatively accelerated observers by resorting to the relative entropy of coherence. We show that the relative entropy of coherence monotonically decreases when acceleration goes up, as a consequence of the Unruh effect. Specifically, the initial states with parameters α=bα=bα=b and α=1−b2α=\sqrt{1-b^2}α=1−b2​ have the same initial relative entropy coherence at a=0a=0a=0 (with aaa the acceleration), but degrade along two different trajectories. The relative entropy of coherence reaches vanishing value in the scalar field in the infinite acceleration limit, but non-vanishing value in the Dirac field. This suggests that in the Dirac field, the bipartite state possesses quantum coherence to some extent with the variation of the relative acceleration, and may lead to potential applications in quantum computation performed by observers in motion relatively.
The bipartite and tripartite entanglement of a 3-qubit fermionic system when one or two subsystems accelerate are investigated. It is shown that all the one-tangles decrease as the acceleration increases. However, unlike the scalar case, here one-tangles NCI(ABI){\mathcal{N}}_{{C}_{I}({\mathrm{AB}}_{I})}NCI​(ABI​)​ and NCI(AB){\mathcal{N}}_{{C}_{I}(\mathrm{AB})}NCI​(AB)​ never reduce to zero for any acceleration. It is found that the system has only tripartite entanglement when either one or two subsystems accelerate, which means that the acceleration does not generate bipartite entanglement and does not affect the entanglement structure of the quantum states in this system. It is of interest to note that the $\ensuremath{\pi}$-tangle of the two-observer-accelerated case decreases much quicker than that of the one-observer-accelerated case and it reduces to a nonzero minimum in the infinite-acceleration limit. Thus we argue that the qutrit systems are better than qubit systems in performing quantum information processing tasks in noninertial systems.
Entanglement degradation caused by the Unruh effect is discussed for the tripartite GHZ or W states constructed by modes of a non-interacting quantum field viewed by one inertial observer and two uniformly accelerated observers. For fermionic states, the Unruh effect even for infinite accelerations cannot completely remove the entanglement. However, for the bosonic states, the situation is different and the entanglement vanishes asymptotically. Also, the entanglement is studied for the bipartite subsystems. While for the GHZ states all the bipartite subsystems are identically disentangled, for the W states the bipartite subsystems are somewhat entangled, though, this entanglement can be removed for appropriately accelerated observers. Interestingly, logarithmic negativity as a measure for determining the entanglement of one part of the system relative to the other two parts, is not generally the same for different parts. This means that we encounter tripartite systems where each part is differently entangled to the other two parts.
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.
We study the entanglement of families of Unruh modes in the Bell states $|{\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rangle}=1/\sqrt{2}(|00\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|11\ensuremath{\rangle})$ and $|{\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rangle}=1/\sqrt{2}(|01\ensuremath{\rangle}\ifmmode\pm\else\textpm\fi{}|10\ensuremath{\rangle})$ shared by two accelerated observers and find fundamental differences in the robustness of entanglement against acceleration for these states. States ${\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}$ are entangled for all finite accelerations, whereas, due to the Unruh effect, states ${\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}$ lose their entanglement for finite accelerations. This is true for Bell states of two bosonic modes, as well as for Bell states of a bosonic and a fermionic mode. Furthermore, there are also differences in the degradation of entanglement for Bell states of fermionic modes. We reveal the origin of these distinct characteristics of entanglement degradation and discuss the role that is played by particle statistics. Our studies suggest that the behavior of entanglement in accelerated frames strongly depends on the occupation patterns of the constituent states, whose superposition constitutes the entangled state, where especially states ${\mathrm{\ensuremath{\Phi}}}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\mathrm{\ensuremath{\Psi}}}^{\ifmmode\pm\else\textpm\fi{}}$ exhibit distinct characteristics regarding entanglement degradation. Finally, we point out possible implications of hovering over a black hole for these states.