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arXiv:1205.1268 · Physical Review A

Vanishing geometric discord in noninertial frames

Eric G. Brown, K. J. R. Cormier, Eduardo Martín-Martínez, Robert B. Mann

Published Sep 10, 2012• 40 citations

Abstract

We revisit the ``field entanglement in noninertial frames'' scenario extensively studied in previous literature and demonstrate that the geometric discord, a proposed measure of generalized quantum correlations, decays to zero in the infinite acceleration limit. This is in contrast with previous research showing that the acceleration-induced degradation of quantum discord was not strong enough to extinguish discord in this limit. We argue that our finding has two opposing but noncontradicting implications. On the one hand, the usable quantum correlations in the large acceleration regime appear severely limited for any protocols making use of geometric discord as a figure of merit. On the other hand, our result indicates, in corroboration with other recent work, that the geometric discord is not a faithful measure of quantum correlations, especially in the case of continuous-variable systems, due to the existence of states which have finite quantum discord but are nevertheless infinitesimally close (in the Hilbert-Schmidt norm) to a zero-discord state.

DOI: 10.1103/physreva.86.032108

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We revisit the ``field entanglement in noninertial frames'' scenario extensively studied in previous literature and demonstrate that the geometric discord, a proposed measure of generalized quantum correlations, decays to zero in the infinite acceleration limit. This is in contrast with previous research showing that the acceleration-induced degradation of quantum discord was not strong enough to extinguish discord in this limit. We argue that our finding has two opposing but noncontradicting implications. On the one hand, the usable quantum correlations in the large acceleration regime appear severely limited for any protocols making use of geometric discord as a figure of merit. On the other hand, our result indicates, in corroboration with other recent work, that the geometric discord is not a faithful measure of quantum correlations, especially in the case of continuous-variable systems, due to the existence of states which have finite quantum discord but are nevertheless infinitesimally close (in the Hilbert-Schmidt norm) to a zero-discord state.

Abstract 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 D G . We notice that the quantum discord still exists for the system ρ ARI even in the acceleration limit. We also calculate the entanglement of formation (EOF) and the geometric measure of entanglement ( E G ) 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.

We calculate the quantum discord between two free modes of a scalar field, which start in a maximally entangled state and then undergo a relative constant acceleration. In a regime where there is no distillable entanglement due to the Unruh effect, we show that there is a finite amount of quantum discord, which is a measure of purely quantum correlations in a state over and above quantum entanglement. Even in the limit of infinite acceleration of the observer detecting one of the modes, we provide evidence for a nonzero amount of purely quantum correlations. We discuss our result in the context of secure quantum communications involving eavesdroppers in noninertial frames.

We study quantum discord, in addition to entanglement, of bipartite pseudo-entanglement in noninertial frames. It is shown that the entanglement degrades from its maximum value in a stationary frame to a minimum value in an infinite accelerating frame. There is a critical region found in which, for particular cases, entanglement of states vanishes for certain accelerations. The quantum discord of pseudo-entanglement decreases by increasing the acceleration. Also, for a physically inaccessible region, entanglement and nonclassical correlation are evaluated and shown to match the corresponding values of the physically accessible region for an infinite acceleration.

We investigate the geometric measure of quantum discord of all possible bipartite divisions of a tripartite system of Dirac fields in noninertial frames. As a comparison, we calculate the geometric measure of entanglement. We discuss the properties of geometric measure of quantum discord and geometric measure of entanglement for three qubit–qubit subsystems with acceleration parameter and the parameter describing the degree of entanglement the system in detail. We have found a conservative relationship involving two of three geometric discords in some condition and another conservative relationship involving three geometric discords for initially maximally entangled states. By the way, we also report some conservative relationships of concurrence, mutual information and geometric measure of entanglement for two bipartite subsystems.

It has recently been pointed out that the geometric quantum discord, as defined by the Hilbert-Schmidt norm (2-norm), is not a good measure of quantum correlations, since it may increase under local reversible operations on the unmeasured subsystem. Here, we revisit the geometric discord by considering general Schatten

p

-norms, explicitly showing that the 1-norm is the only

p

-norm able to define a consistent quantum correlation measure. In addition, by restricting the optimization to the tetrahedron of two-qubit Bell-diagonal states, we provide an analytical expression for the 1-norm geometric discord, which turns out to be equivalent to the negativity of quantumness. We illustrate the measure by analyzing its monotonicity properties.

In this article, we investigate the geometric discord of quantum states of fermionic system in accelerated frame. It is shown by the method beyond the single-mode approximation, depending on the region considered, that the geometric discord for the entangled quantum states of fermionic system in accelerated frame can vanish or be retained at the infinite acceleration limit: it does not disappear when the quantum state of the particle(Alice)-particle(Bob in region I) case or the particle(Alice)-antiparticle(Bob in region II) is considered and it disappears when the particle(Alice)-antiparticle(Bob in region I) case or the particle(Alice)-particle(Bob in region II) one is considered.