Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts

We consider a quantum oscillator coupled to a bath of $N$ other oscillators. The total system evolves with a quasi-Hermitian Hamiltonian. Associated to it is a family of Hermitian systems, parameterized by a unitary map $W$. Our main goal is to find the influence of $W$ on the entropy and the entanglement in the Hermitian systems. We calculate explicitly the reduced density matrix of the single oscillator for all Hermitian systems and show that, regardless of $W$, their von Neumann entropy oscillates with a common period which is twice that of the non-Hermitian system. We show that, generically, the oscillator and the bath are entangled for almost all times. While the amount of entanglement depends on the choice of $W$, the entanglement of the time-averaged density matrix is entirely independent of $W$. These results describe some universality in the physical properties of all Hermitian systems associated to a given non-Hermitian one.

Figure 1

Parameters $Q_0=0.5, \mathrm{\Delta }\equiv {max}_t\mathrm{\Delta }\left(t\right)=0.2$. According to (60), $Q\left(t\right)$ starts at $Q_m=0.3$ (at a time we take to be $t=0$) and moves to $Q_M=0.7$ at time $\omega t=\pi /2$, and then back to $Q_m$ at time $\omega t=\pi$ (top panel), so the value of the entropy $\mathcal{E}\left(Q\right(t\left)\right)$ evolves through two periods (bottom panel). In each period, the entropy has two local minima (counting minima at the endpoints of the considered intervals once).

Figure 2

Parameters $Q_0=0.6, \mathrm{\Delta }={max}_t\mathrm{\Delta }\left(t\right)=0.2$. In this case $Q\left(t\right)$ starts at $Q_m=0.4$ when $t=0$ (upon a possible shift of the time axis) and moves to $Q_M=0.8$ at time $\omega t=\pi /2$ and back to $Q_m$ at time $\omega t=\pi$ (top panel). The value of the entropy $\mathcal{E}\left(Q\right(t\left)\right)$ evolves through one single period (bottom panel). In each period, the entropy has two local minima (counting minima at the endpoints once).

Figure 3

Parameters $Q_0=0.8, \mathrm{\Delta }={max}_t\mathrm{\Delta }\left(t\right)=0.2$. In this case $Q\left(t\right)$ starts at $Q_m=0.6$ when $t=0$ (after possibly shifting the time axis) and moves to $Q_M=1.0$ at time $\omega t=\pi /2$ and back to $Q_m$ at time $\omega t=\pi$ (top panel). The value of the entropy $\mathcal{E}\left(Q\right(t\left)\right)$ evolves through one single period (bottom panel). Since 0.5 is not in the interval $(Q_m,Q_M)$, the graph of the entropy has only one local minimum in each period, instead of two when the interval contains the value 0.5 for $Q$.

Figure 4

Comparing the entropies $\mathcal{E}\left({\overline{\rho }}_{h_W}\left(t\right)\right)$ (solid line, online in green) and $\mathcal{E}\left({\overline{\rho }}_H\left(t\right)\right)$ (dashed line, online in red) for the values $c=0.5$ and $\alpha =0,0.15,0.3,0.45$. The doubling of the period for $\alpha \ne 0$ is manifest. The oscillations decrease as $\alpha$ approaches 0.5, which gives the stationary state.