Sudden death of entanglement at finite temperature

We consider the decay of quantum entanglement quantified by the concurrence of a pair of two-level systems, each of which is interacting with a reservoir at finite temperature $T$. For a broad class of initially entangled states, we demonstrate that the system always becomes disentangled in a finite time, i.e., “entanglement sudden death” occurs. This class includes all states which previously had been found to have long-lived entanglement in zero-temperature reservoirs. Our general result is illustrated by an example.

Figure 1

Disentanglement by spontaneous emission of a two-qubit system in a heat bath. The reservoir is modeled by different harmonic oscillator modes. Each qubit, here depicted by a two-level atom, interacts with its reservoir. The only interaction between the qubits that ever exists is the one that leads to their entanglement at time $t=0$. Following this, however, the only interaction that remains is that with the corresponding reservoir. This leads to decoherence, which causes the qubits to disentangle. Here we include the effect of heat by studying the system at $T\ne 0$.

Figure 2

(Color online) Plot of $F\left(x\right)$ vs $X$. This is the plot of the first quartic equation in (8) for $\overline{n}=0.8$, $a=0.1$, $d=0.05$, and $z=0.3$.

Figure 3

(Color online) Plot of $C$ (concurrence) vs $X$ $\left(=e^{-\Gamma \left(2\overline{n}+1\right)t}\right)$ vs $\alpha$. $C=0$ corresponds to no entanglement. $X=1$ corresponds to $t=0$, while $X=1$ corresponds to $t=\infty$. Notice that as soon as $\overline{n}$ becomes finite, for all values of $\alpha$, $C$ becomes zero at $X<0$, i.e., entanglement decays in a finite time. As $\overline{n}$ becomes larger, all states disentangle at approximately $X=0.5$.