Entanglement of Dirac fields in noninertial frames

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.

Figure 1

Rindler spacetime diagram: lines of constant position $\zeta$ are hyperbolas and lines of constant proper time $\tau$ for the accelerated observer run through the origin. Note that while $\tau$ flows in the direction of $t$ in region I, it flows in the direction of $-t$ in region II (i.e., the dashed line rotates counterclockwise for increasing values of $\tau$). A uniformly accelerated observer Rob $\left(R\right)$ with acceleration $a$ travels on a hyperbola constrained to region I, while a fictitious observer anti-Rob $\left(\overline{R}\right)$ travels on a corresponding hyperbola in region II given by the negative of Rob’s coordinates. The horizons $H_{\pm }$ are lines of $\tau =\pm \infty$ which Alice $\left(A\right)$ will cross at finite Minkowski times.

Figure 2

(Color online) Bipartite pure-state entanglement. Thick solid curve (blue): anti-Rob in region II with (Alice and Rob in region I). Thin dashed curve (red): Alice with (regions I and II). Dotted curve (green): Rob in region I with (Alice and anti-Rob region II).

Figure 3

(Color online) The logarithmic negativity as a function of $r$. Thick solid curve (blue): between Alice and Rob in region I. Thin dashed curve (red): between the modes in regions I and II. Dotted curve (green): between Alice and anti-Rob in region II.

Figure 4

(Color online) Entanglement of formation as a function of $r$. Thick solid curve (blue): between Alice and Rob in region I. Thin dashed curve (red): between the modes in regions I and II. Dotted curve (green): between Alice and anti-Rob in region II.

Figure 5

(Color online) Mutual information as a function of $r$. Thick solid curve (blue): between Alice and Rob in region I. Thin dashed curve (red): between the modes in regions I and II. Dotted curve (green): between Alice and anti-Rob in region II.

Figure 6

(Color online) Bipartite tangles as a function of $r$. Thick solid curve (blue): between Alice and Rob in region I. Thin dashed curve (red): between the modes in regions I and II. Dotted curve (green): between Alice and anti-Rob in region II.

Figure 7

(Color online) Separable uncertainties as a function of $r$. Thick solid curve (blue): between Alice and Rob in region I. Thin dashed curve (red): between the modes in regions I and II. Dotted curve (green): between Alice and anti-Rob in region II.

Figure 8

(Color online) Single-qubit properties as a function of $r$. Dotted-solid line (blue-green): Alice (always zero). Thick-thin dashed curve (blue-red): Rob in region I. Dot-dashed curve (green-red): anti-Rob in region II.