Average entanglement dynamics in open two-qubit systems with continuous monitoring

We present a comprehensive implementation of the quantum trajectory theory for the description of the entanglement dynamics in a Markovian open quantum system made of two qubits. We introduce the average concurrence to characterize the entanglement in the system and derive a deterministic evolution equation for it that depends on the ways in which information is read from the environment. This buildt-in flexibility of the method is used to address two actual issues in quantum information: entanglement protection and entanglement estimation. We identify general physical situations in which an entanglement protection protocol based on local monitoring of the environment can be implemented. Additionally, we methodically find unravelings of the system dynamics providing analytical tight bounds for the unmonitored entanglement in the system at all times. We conclude by showing the independence of the method from the choice of entanglement measure.

Figure 1

Entanglement protection for a $2\times 2$ system with its first qubit incoherently coupled to a phase damping environment and its second to a thermal bath at infinite temperature. We assume that the coupling strength to both baths lead to the same decay rate $\gamma$. Initially the system is prepared in the entangled pure state $|\psi \left(0\right)\rangle =\left(\right|00\rangle -|01\rangle +i|10\rangle +i\sqrt{5}|11\rangle )/\sqrt{8}$ with $C_0=0.809$. If the local unraveling $u^p$ is implemented for the continuous monitoring of each of the environments, then the entanglement in the system is protected along each single quantum trajectory, the concurrence distribution becomes localized, and the average concurrence $C_{u^p}$ remains constant for all times (solid line). If, however, the protection is only partial, implementing $u^p$ for the monitoring of the phase damping channel only, while $u=\left(\begin{array}{cc}\hfill -1& 0\\ \hfill 0& 1\end{array}\right)$ is used for the thermal bath, the average concurrence decays exponentially with $C_u\left(t\right)=C_0{e}^{-2\gamma t}$ (dashed line), but still more slowly than the unmonitored entanglement. Additionally, concurrence fluctuates randomly on the ensemble of trajectories. At each time the probability density distribution of concurrence is depicted in the background (darker regions correspond to higher densities). For times shorter than the characteristic time of the system entanglement $\sim 1/2\gamma$, most of the trajectories exhibit an increase in their concurrence, i.e., entanglement creation, a situation that is reversed for longer times when states in most of the trajectories become separable.

Figure 2

Concurrence dynamical upper-bound performance. The entanglement evolution of an ensemble of $10000$ initially pure states of a $2\times 2$ system coupled to phase damping channels is depicted for two times. In the upper panels, points indicate the concurrence of the final-state $C\left(\rho \right)$ in units of the upper bound $C_{u^{+}}$ versus the initial-state concurrence $C_0$. Darker regions contain more data points. Lower panels display the cumulative density function (CDF) (solid blue line) corresponding to the probability density function in the upper panels. The CDF corresponding to the exponential bound (dashed black line) is also shown for comparison.

Figure 3

Concurrence-of-assistance dynamical lower-bound performance. The entanglement evolution of an ensemble of $10000$ initially pure states of a $2\times 2$ system coupled to phase damping channels is depicted for two times. In the upper panels, points indicate the concurrence of assistance of the final-state $C_A\left(\rho \right)$ in units of the lower bound $C_{u^{-}}$ versus the initial-state concurrence $C_0$. Darker regions contain more data points. Lower panels display the cumulative density function (CDF) (solid blue line) corresponding to the probability density function in the upper panels. The CDF corresponding to the constant—entanglement protecting—bound (dashed black line) is also shown for comparison.

Figure 4

Concurrence-of-assistance dynamical lower-bound performance. The entanglement evolution of an ensemble of $10000$ initially pure states of a $2\times 2$ system coupled to amplitude damping baths is depicted for two times. In the upper panels the points indicate the concurrence of assistance of the final-state $C_A\left(\rho \right)$ in units of the lower bound $C_{u^{-}}$ versus the initial-state concurrence $C_0$. Darker regions contain more data points. Lower panels display the cumulative density function (CDF) (solid blue line) corresponding to the probability density function in the upper panels. The CDF corresponding to the exponential bound in Eq. (24) (dashed black line) is also shown for comparison.