Unified entropies and quantum speed limits for nonunitary dynamics

We discuss a class of quantum speed limits (QSLs) based on unified quantum ($\alpha ,\mu$)-entropy for nonunitary physical processes. The bounds depend on both the Schatten speed and the smallest eigenvalue of the evolved state and the two-parametric unified entropy. We specialize these results to quantum channels and non-Hermitian evolutions. In the first case, the QSL depends on the Kraus operators related to the quantum channel, while in the second case the speed limit is recast in terms of the non-Hermitian Hamiltonian. To illustrate these findings, we consider a single-qubit state evolving under the (i) amplitude damping channel and (ii) the nonunitary dynamics generated by a parity-time-reversal symmetric non-Hermitian Hamiltonian. The QSL is nonzero at earlier times, while it becomes loose as the smallest eigenvalue of the evolved state approaches zero. Furthermore, we investigate the interplay between unified entropies and speed limits for the reduced dynamics of quantum many-body systems. The unified entropy is upper bounded by the quantum fluctuations of the Hamiltonian of the system, while the QSL is nonzero when entanglement is created by the nonunitary evolution. Finally, we apply these results to the XXZ model and verify that the QSL asymptotically decreases as the system size increases. Our results find applications to nonequilibrium thermodynamics, quantum metrology, and equilibration of many-body systems.

Figure 1

Phase diagram for the function ${\left[f_{\alpha }\left(\rho \right)\right]}^{\mu -1}$, with $\rho \in \mathcal{S}$. On the one hand, the $\alpha$-purity satisfies the inequality $f_{\alpha }\left(\rho \right)\ge 1$ for $\alpha \in [0,1]$, and one finds ${\left[f_{\alpha }\left(\rho \right)\right]}^{\mu -1}\le 1$ for $0<\mu <1$, while ${\left[f_{\alpha }\left(\rho \right)\right]}^{\mu -1}\ge 1$ for $\mu >1$. On the other hand, the $\alpha$-purity behaves as $f_{\alpha }\left(\rho \right)\le 1$ for $\alpha \ge 1$, which implies that ${\left[f_{\alpha }\left(\rho \right)\right]}^{\mu -1}\ge 1$ for $0<\mu <1$, while we have that ${\left[f_{\alpha }\left(\rho \right)\right]}^{\mu -1}\le 1$ for $\mu >1$.

Figure 2

Density plot of QSL time ${\tau }_{\alpha ,\mu }^{\text{QSL}}$ [see Eq. (15)] and the normalized relative error ${\tilde{\delta }}_{\alpha ,\mu }\left(\tau \right)$ [see Eq. (13)], as a function of the dimensionless parameter $\gamma \tau$, for a single-qubit state evolving under the amplitude damping channel. Here we choose the initial state ${\rho }_0=(1/2)(\mathbb{I}+\vec{r}·\vec{\sigma })$ with $\{r,\theta \}=\{1/2,\pi /4,\pi /4\}$.

Figure 3

Plot of the QSL time ${\tau }_{\alpha ,0}^{\text{QSL}}$ [see Eq. (17)] and the relative error ${\tilde{\delta }}_{\alpha ,0}\left(\tau \right)$ [see Eq. (13)], for the non-Hermitian two-level system described by the Hamiltonian $\tilde{H}=\varpi {\sigma }_x+i\eta {\sigma }_z$. The system is initialized at the single-qubit state ${\rho }_0=(1/2)(\mathbb{I}+\vec{r}·\vec{\sigma })$, with $\{r,\theta ,\varphi \}=\{0.5,\pi /4,\pi /4\}$. Here we set the cases $\eta /\varpi =0.5$ [panels (a), (d)], $\eta /\varpi =1$ [panels (b), (e)], and $\eta /\varpi =2$ [panels (c), (f)].

Figure 4

Plot of the smallest eigenvalue ${\kappa }_{\text{min}}\left({\rho }_{\tau }\right)$ as a function of the dimensionless parameter $\varpi \tau$. Here ${\rho }_{\tau }=e^{-i\tau \tilde{H}}{\rho }_0{e}^{+i\tau {\tilde{H}}^{†}}/\text{Tr}\left(e^{-i\tau \tilde{H}}{\rho }_0{e}^{+i\tau {\tilde{H}}^{†}}\right)$ is the evolved state of the system, with $\tilde{H}=\varpi {\sigma }_x+i\eta {\sigma }_z$ being a two-level non-Hermitian Hamiltonian. We set the initial single-qubit state ${\rho }_0=(1/2)(\mathbb{I}+\vec{r}·\vec{\sigma })$, with $\{r,\theta ,\varphi \}=\{0.5,\pi /4,\pi /4\}$. Here we plot the cases $\eta /\varpi =0.5$ (black dashed line), $\eta /\varpi =1$ (blue dot dashed line), and $\eta /\varpi =2$ (red solid line).

Figure 5

Plot of the QSL time ${\tau }_{\alpha ,0}^{\text{QSL}}$ [see Eq. (24)] and the relative error ${\tilde{\delta }}_{\alpha ,0}\left(\tau \right)$ [see Eq. (25)], for the XXZ model, with $J\mathrm{\Delta }=0.5$ [see Eq. (26)]. The system is initialized at the state ${\rho }_0=((1-p)/d)\mathbb{I}+p|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }|$, with $d=2^L$, where $|\mathrm{\Psi }\rangle =|1,0,1,0,...,0,1\rangle$ is the Néel state, while $|0\rangle$ and $|1\rangle$ stand for the spin-up and -down state, respectively. Here we set the system sizes $L=\{2,4,6\}$, with open boundary conditions, while $L_A=1$, and $L_B=\{1,3,5\}$, also fixing the mixing parameter $p=0.5$.