Mixedness timescale in non-Hermitian quantum systems

We discuss the short-time perturbative expansion of the linear entropy for finite-dimensional quantum systems whose dynamics can be effectively described by a non-Hermitian Hamiltonian. We derive a timescale for the degree of mixedness for an input state undergoing non-Hermitian dynamics and specialize these results in the case of a driven-dissipative two-level system. Next, we derive a timescale for the growth of mixedness for bipartite quantum systems that depends on the effective non-Hermitian Hamiltonian. In the Hermitian limit, this result recovers the perturbative expansion for coherence loss in Hermitian systems, while it provides an entanglement timescale for initial pure and uncorrelated states. To illustrate these findings, we consider the many-body transverse-field XY Hamiltonian coupled to an imaginary all-to-all Ising model. We find that the non-Hermitian Hamiltonian enhances the short-time dynamics of the linear entropy for the considered input states. Overall, each timescale depends on minimal ingredients such as the probe state and the non-Hermitian Hamiltonian of the system, and its evaluation requires low computational cost. Our results find applications to non-Hermitian quantum sensing, quantum thermodynamics of non-Hermitian systems, and $\mathcal{PT}$-symmetric quantum field theory.

Figure 1

Density plot of the dimensionless quantities $1/\gamma T_1$ [see Eq. (8)] and $1/{\gamma }^2{T}_2^2$ [see Eq. (9)], as a function of the mixing parameter $r\in [0,1]$ and the azimuthal angle $\theta \in [0,\pi ]$, respective to the perturbative expansion of the linear entropy for the driven dissipative two-level system around [see Eq. (3)]. In panels (b)–(d) we set the polar angle $\varphi =\pi /4$ and consider the ratio $\mathrm{\Omega }/\gamma =0.1$ (b), $\mathrm{\Omega }/\gamma =1$ (c), and $\mathrm{\Omega }/\gamma =0.1$ (d).

Figure 2

Plot of the linear entropy $S_L\left({\tilde{\rho }}_t\right)$, as a function of the dimensionless parameter $\gamma t$, for the driven two-level system described by the Hamiltonian $H=\mathrm{\Delta }|1\rangle \langle 1|+(\mathrm{\Omega }/2)\left(\right|0\rangle \langle 1|+|1\rangle \langle 0\left|\right)$. Here we choose the ratio $\mathrm{\Delta }/\gamma =0.5$ and set $\mathrm{\Omega }/\gamma =0.1$ (a, d), $\mathrm{\Omega }/\gamma =1$ (b, e), and $\mathrm{\Omega }/\gamma =10$ (c, f). The system is initialized in the single-qubit state ${\rho }_0=(1/2)(\mathbb{I}+\vec{r}·\vec{\sigma })$, with $\{r,\theta ,\varphi \}=\{1/4,\pi /4,\pi /4\}$ (a–c) and $\{r,\theta ,\varphi \}=\{1/4,3\pi /4,\pi /4\}$ (d–f). The black dotted line indicates the linear entropy $S_L\left({\rho }_0\right)=[d/(d-1)][1-\text{Tr}\left({\rho }_0^2\right)]$ respective to the initial state. The cyan dash-dotted line depicts the linear entropy $S_L\left({\rho }_t\right)=[d/(d-1)][1-\text{Tr}\left({\rho }_t^2\right)]$, where ${\rho }_t$ satisfies the Markovian master equation in Eq. (7). The blue solid line indicates the linear entropy $S_L\left({\tilde{\rho }}_t\right)=[d/(d-1)][1-\text{Tr}\left({\tilde{\rho }}_t^2\right)]$, where the normalized state ${\tilde{\rho }}_t={\rho }_t/\text{Tr}\left({\rho }_t\right)$ fulfills Eq. (2), with $H_1=H$ and $H_2=-(\gamma /2)|1\rangle \langle 1|$. The red dashed line represents the linear entropy $S_L\left({\tilde{\rho }}_t\right)\approx S_L\left({\rho }_0\right)-[d/(d-1)](t/T_1+t^2/T_2^2)$ within the short-time approximation [see Eqs. (8) and (9)].

Figure 3

Plot of the linear entropy $S_L\left({\tilde{\rho }}_t^A\right)$ for the $k$-particle reduced density matrix ${\tilde{\rho }}_t^A$, as a function of the dimensionless parameter $Jt$. The nonunitary evolution of subsystem $A$ is governed by Eq. (10), with $H_1$ being the transverse field XY Hamiltonian in Eq. (20), and $H_2$ as the all-to-all Ising model in Eq. (21). The system $A+B$ is initialized in the GHZ mixed state ${\rho }_0^{AB}=((1-p)/d)\mathbb{I}+p|{\text{GHZ}}_N\rangle \langle {\text{GHZ}}_N|$, where $|{\text{GHZ}}_N\rangle =(1/\sqrt{2})({|0\rangle }^{\otimes N}+{|1\rangle }^{\otimes N})$. Here we set $N=8$, $\gamma =0.75$, $J_z/J=0.5$, and the mixing parameter $p=0.5$. The blue solid line corresponds to the exact expression of linear entropy in Eq. (11), and the red dashed line indicates its the short-time perturbative expansion in Eq. (12) [see also Eqs. (24), (25), and (26)].

Figure 4

Plot of the linear entropy $S_L\left({\tilde{\rho }}_t^A\right)$ in the short-time approximation [see Eq. (12)] for the $k$-particle reduced density matrix ${\tilde{\rho }}_t^A$, as a function of the dimensionless parameter $Jt$. The nonunitary evolution of subsystem $A$ is governed by Eq. (10), with $H_1$ being the transverse field XY Hamiltonian in Eq. (20) and $H_2$ as the all-to-all Ising model in Eq. (21). The system $A+B$ is initialized in the GHZ mixed state ${\rho }_0^{AB}=((1-p)/d)\mathbb{I}+p|{\text{GHZ}}_N\rangle \langle {\text{GHZ}}_N|$, where $|{\text{GHZ}}_N\rangle =(1/\sqrt{2})({|0\rangle }^{\otimes N}+{|1\rangle }^{\otimes N})$. Here we set $N=8$, $\gamma =0.75$, and the mixing parameter $p=0.5$. The blue solid line corresponds to the case $J_z/J=0.5$, and the red dashed line depicts the case $J_z/J=0$.

Figure 5

Plot of the linear entropy $S_L\left({\tilde{\rho }}_t^A\right)$ [see Eq. (11)] in the short-time approximation [see Eq. (12)], for the $k$-particle reduced density matrix ${\tilde{\rho }}_t^A$, as a function of the dimensionless parameter $Jt$. The nonunitary evolution of subsystem $A$ is governed by Eq. (10), with $H_1$ being the transverse field XY Hamiltonian in Eq. (20), and $H_2$ as the all-to-all Ising model in Eq. (21). The system $A+B$ is initialized in the GHZ mixed state ${\rho }_0^{AB}=[(1-p)/d]\mathbb{I}+p|{\text{GHZ}}_N\rangle \langle {\text{GHZ}}_N|$, where $|{\text{GHZ}}_N\rangle =(1/\sqrt{2})({|0\rangle }^{\otimes N}+{|1\rangle }^{\otimes N})$. Here we consider the system size $N=8$, with $k=5$, and $\gamma =0.75$. The blue solid line corresponds to the case $J_z/J=0.5$, and the red dashed line depicts the case $J_z/J=0$.

Figure 6

Plot of the first-order time derivative $-{[d{S}_L(•)/dt]}_{t=0}$ around $t=0$ of the linear entropy, and the timescale $1/\gamma T_1$, as a function of the Bloch sphere radius $r$, for the driven two-level system described by the Hamiltonian $H=\mathrm{\Delta }|1\rangle \langle 1|+(\mathrm{\Omega }/2)\left(\right|0\rangle \langle 1|+|1\rangle \langle 0\left|\right)$. Here we choose the ratio $\mathrm{\Delta }/\gamma =0.5$, and $\mathrm{\Omega }/\gamma =0.1$. The system is initialized in the single-qubit state ${\rho }_0=(1/2)(\mathbb{I}+\vec{r}·\vec{\sigma })$, with $\{\theta ,\varphi \}=\{3\pi /4,\pi /4\}$. The black dotted line indicates $-{[d{S}_L\left({\rho }_0\right)/dt]}_{t=0}$, with $S_L\left(e^{-itH}{\rho }_0{e}^{+itH}\right)=S_L\left({\rho }_0\right)$. The cyan dash-dotted line depicts the quantity $-{[d{S}_L\left({\rho }_t\right)/dt]}_{t=0}$, with the state ${\rho }_t$ evolving under the Markovian master equation in Eq. (7). The blue solid line indicates the first-order time derivative $-{[d{S}_L\left({\tilde{\rho }}_t\right)/dt]}_{t=0}$, with the normalized state ${\tilde{\rho }}_t={\rho }_t/\text{Tr}\left({\rho }_t\right)$ satisfying Eq. (2), where $H_1=H$ and $H_2=-(\gamma /2)|1\rangle \langle 1|$. The red dashed line displays the timescale $1/\left(\gamma T_1\right)$ [see Eq. (8)].

Figure 7

Plot of the second-order time derivative $-(1/2){[d^2{S}_L(•)/d{t}^2]}_{t=0}$ around $t=0$ of the linear entropy, and the timescale $1/{\gamma }^2{T}_2^2$, as a function of the radius $r$ of the Bloch sphere, for the driven two-level system described by the Hamiltonian $H=\mathrm{\Delta }|1\rangle \langle 1|+(\mathrm{\Omega }/2)\left(\right|0\rangle \langle 1|+|1\rangle \langle 0\left|\right)$. The system is initialized in the single-qubit state ${\rho }_0=(1/2)(\mathbb{I}+\vec{r}·\vec{\sigma })$, with $\{\theta ,\varphi \}=\{3\pi /4,\pi /4\}$. We set $\mathrm{\Omega }/\gamma =0.1$ (a), $\mathrm{\Omega }/\gamma =1$ (b), and $\mathrm{\Omega }/\gamma =10$ (c). The black dotted line shows the second-order time derivative $-(1/2){[d^2{S}_L\left({\rho }_0\right)/d{t}^2]}_{t=0}$. The cyan dash-dotted line depicts the quantity $-(1/2){[d^2{S}_L\left({\rho }_t\right)/d{t}^2]}_{t=0}$, where the state ${\rho }_t$ fulfills Eq. (7). The blue solid line indicates the quantity $-(1/2){[d^2{S}_L\left({\tilde{\rho }}_t\right)/d{t}^2]}_{t=0}$, with the normalized state ${\tilde{\rho }}_t={\rho }_t/\text{Tr}\left({\rho }_t\right)$ satisfying Eq. (2), where $H_1=H$ and $H_2=-(\gamma /2)|1\rangle \langle 1|$. The red dashed line displays the timescale $1/\left({\gamma }^2{T}_2^2\right)$ [see Eq. (9)].