Increase of entanglement by local $\mathcal{PT}$-symmetric operations

Entanglement plays a central role in the field of quantum information science. It is well known that the degree of entanglement cannot be increased under local operations. Here, we show that the concurrence of a bipartite entangled state can be increased under the local $\mathcal{PT}$-symmetric operation. This violates the property of entanglement monotonicity. We also use the Bell–Clauser-Horne-Shimony-Holt and steering inequalities to explore this phenomenon.

Figure 1

The schematic diagram of the dynamics of the two qubits. Qubit 1 undergoes either the Rabi process or a local $\mathcal{PT}$-symmetric operation, while qubit 2 remains isolated. The initial condition is the maximally entangled state: $\left(\right|00\rangle +|11\rangle )/\sqrt{2}$.

Figure 2

The dynamics of (a) the concurrence $\mathcal{C}$, (b) the maximal mean value of the Bell kernel $\langle {\mathcal{B}}_{\text{max}}\rangle$, and (c) the steering parameter $S_3$. The black solid curve in each figure represents that qubit 1 undergoes the coherent Rabi oscillation, while the red dotted curve represents that the local $\mathcal{PT}$-symmetric operation is performed on qubit 1. The blue dashed lines in (b) and (c) are the classical bounds [i.e., upper bounds in Eqs. (12) and (13)] of $\langle {\mathcal{B}}_{\text{max}}\rangle$ and $S_3$, respectively. In plotting the figure, the time $t$ is in units of $1/\Delta E$, and the initial condition is $\left(\right|00\rangle +|11\rangle )/\sqrt{2}$.

Figure 3

The schematic diagram of the dynamics of the two qubits. Qubit 1 is subjected to an amplitude damping for a time $t_c$. At the time $t_c$, the damping is turned off, and the local $\mathcal{PT}$-symmetric operation is then performed on qubit 1. The initial state of the system is $\left(\right|00\rangle +|11\rangle )/\sqrt{2}$.

Figure 4

The dynamics of (a) the concurrence $\mathcal{C}$, (b) the maximal mean value of the Bell kernel $\langle {\mathcal{B}}_{\text{max}}\rangle$, and (c) the steering parameter $S_3$ when the local $\mathcal{PT}$-symmetric operation is performed on qubit 1. The black dashed, red solid, and blue dotted curves represent the results of different initial states of the $\mathcal{PT}$-symmetric evolution given by Eqs. (18), (19), and (20), respectively. The green horizontal lines in (b) and (c) are the classical bounds [i.e., upper bounds in Eqs. (12) and (13)] of $\langle {\mathcal{B}}_{\text{max}}\rangle$ and $S_3$, respectively. In plotting the figure, the time $t$ of the $\mathcal{PT}$-symmetric evolution is in units of $1/\Delta E$.

Figure 5

The concurrence as a function of time when qubit 1 is under the Hamiltonian of Eq. (23). The black dashed, red solid, and blue dotted curves represent the results of the different initial states given by Eqs. (18), (19), and (20) respectively. In plotting the figure, the time $t$ is in units of $1/\Delta E$, and $\epsilon$ is set to 0.01.