Quantum speedup dynamics process without non-Markovianity

Abstract

The Hamiltonian corrections induced by the action of coherent driving forces are often used to fight dissipative and decoherence mechanisms in experiments. For this reason, in the cases of two exactly solvable decoherence models, we propose a scheme for changing the quantum average speed of an open system by controlling an external unitary coherent driving Hamiltonian on the system. For the Markovian dynamics process, with a judicious choice of the coherent driving parameters, the tightness of quantum speed limit of the dynamics process can be weakened. That is to say, we may drive the open system to the quantum speedup evolution. And in the non-Markovian regime, the original quantum speedup dynamics may be brought to the greater degree of speedup by adding the coherent driving. Somewhat contrary to the previous speedup scenarios by the controllable non-Markovianity, the changes in the excited population and coherence caused by the coherent driving are the intrinsic physical reason for quantum speedup in our scheme.

Appendix

Tightness of QSLTs in the amplitude damping channel and the pure dephasing channel: Firstly, we especially point out two notable QSL bounds derived previously. For the nonunitary dynamics between the arbitrary initial state $\rho _{0}$ to its target state $\rho _{\tau }$, in [21, 22] the authors choose the Euclidean distance ${\mathbf {D}}(\rho _{0},\rho _{\tau })=\Vert \rho _{\tau }-\rho _{0}\Vert _{hs}$ to acquire a new bound of QSLT

$$\begin{aligned} \tau ^{M}_{qsl}=\frac{\Vert \rho _{\tau }-\rho _{0}\Vert _{hs}}{\frac{1}{\tau }\int ^{\tau }_{0}\Vert {\dot{\rho }}_{t}\Vert _{hs}\mathrm{d}t}, \end{aligned}$$

(15)

where $\Vert \cdot \Vert _{hs}=\sqrt{\sum ^{d}_{i=1}\sigma ^{2}_{i}}$ being the Hilbert–Schmidt norm, and $\sigma _{i}$ being the singular values of ${\dot{\rho }}_{t}$. By considering the relative purity ${\mathbf {f}}(\tau )=\mathbf {tr}[\rho _{\tau }\rho _{0}]/\mathbf {tr}(\rho ^{2}_{0})$ as a distance measure, we also obtain a bound of the QSLT for arbitrary initially mixed state $\rho _{0}$ to $\rho _{\tau }$ as follows [16] $\tau ^{U}_{qsl}=|{\mathbf {f}}(\tau )-1|\mathbf {tr}(\rho ^{2}_{0}){\cdot }\mathbf {Max}[1/\frac{1}{\tau }\int ^{\tau }_{0}\sum ^{d}_{i=1}\sigma _{i}\varrho _{i}\mathrm{d}t,1/\frac{1}{\tau }\int ^{\tau }_{0}\sqrt{\sum ^{d}_{i=1}\sigma ^{2}_{i}}\mathrm{d}t]$, with $\varrho _{i}$ being the singular values of the initial state $\rho _{0}$. For a pure initial state $\rho _{0}=|\phi _{0}\rangle \langle \phi _{0}|$, the singular value $\varrho _{i}=\delta _{i,1}$, then $\sum ^{d}_{i=1}\sigma _{i}\varrho _{i}=\sigma _{1}\le \sqrt{\sum ^{d}_{i=1}\sigma ^{2}_{i}}$. The bound $\tau _{U}$ can be equal to the bound of the QSLT for a pure initial state obtained in [15], as follows:

$$\begin{aligned} \tau ^{D}_{qsl}=\frac{\sin ^{2}[\arccos (\sqrt{\langle \phi _{0}|\rho _{\tau }|\phi _{0}\rangle })]}{{\frac{1}{\tau }\int ^{\tau }_{0}\Vert {\dot{\rho }}_{t}\Vert _{op}\mathrm{d}t}}, \end{aligned}$$

(16)

where the operator norm $\Vert {\dot{\rho }}_{t}\Vert _{op}=\sigma _{1}$ is the largest singular value of ${\dot{\rho }}_{t}$.

Next, we must ensure the above bounds are fairly compared with each other. It can be clear to find that the dynamics-dependent term that appears in the above bounds is given by $\Vert {\dot{\rho }}_{t}\Vert _{hs}$ and $\Vert {\dot{\rho }}_{t}\Vert _{op}$, respectively. In this work, we mainly consider the quantum evolution in the amplitude damping process and the dephasing noise process. In these two noisy channels, $\Vert {\dot{\rho }}_{t}\Vert _{hs}=\sqrt{2}\Vert {\dot{\rho }}_{t}\Vert _{op}$. This fact allows us to compare the distance term which depend only on the initial and final target states, regardless of the chosen dynamical process. Then, the tighter bound can be clearly acquired.

For the amplitude damping channel, the state of the two-dimension system can evolve to $\rho _{t}=\frac{1}{2}({\mathbb {I}}+r^{t}_{x}\sigma _{x}+r^{t}_{y}\sigma _{y}+r^{t}_{z}\sigma _{z})$, with $r^{t}_{x}=r_{x}\sqrt{p_{t}}$, $r^{t}_{y}=r_{y}\sqrt{p_{t}}$, and $r^{t}_{z}=(1+r_{z})p_{t}-1$. By considering the pure initial state, that is $\Vert r\Vert =\sqrt{r^{2}_{x}+r^{2}_{y}+r^{2}_{z}}=1$. Then, the above bounds can be acquired in the following forms, ${\tau ^{M}_{qsl}}/{\tau }={X}/{\int ^{\tau }_{0}\Vert {\dot{\rho }}_{t}\Vert _{hs}\mathrm{d}t}$ and ${\tau ^{D}_{qsl}}/{\tau }={\sqrt{2}Y}/{\int ^{\tau }_{0}\Vert {\dot{\rho }}_{t}\Vert _{hs}\mathrm{d}t}$, where $X=\sqrt{(1-r^{2}_{z})(\sqrt{p_{\tau }}-1)^{2}+(1+r_{z})^{2}(p_{\tau }-1)^{2}}/\sqrt{2}$, and $Y=|-r_{z}(1+r_{z})(p_{\tau }-1)+(1-r^{2}_{z})(\sqrt{p_{\tau }}-1)|/2$. By using the inequality relationship $|A+B|\le \sqrt{A^{2}+B^{2}}$, and $0{\le }r^{2}_{z}\le 1$, so we can find $\sqrt{2}Y\le \sqrt{r^{2}_{z}(1+r_{z})^{2}(p_{\tau }-1)^{2}+(1-r^{2}_{z})^{2}(\sqrt{p_{\tau }}-1)^{2}}/\sqrt{2}{\le }X$. It is worth noting that ${\tau ^{M}_{qsl}}\ge {\tau ^{D}_{qsl}}$, and this means the bound derived in [21, 22] is much tighter than the bound in [15] for the evolution from an arbitrary pure state to an evolutional target state. Specially, when the qubit is in the initially excited state, that is, $r_{z}=1$, the above two bounds $\tau ^{M}_{qsl}$ and $\tau ^{D}_{qsl}$ are equal to each other.

When we focus on the pure dephasing channel, the state of the qubit evolves as $\rho _{t}=\frac{1}{2}({\mathbb {I}}+r^{t}_{x}\sigma _{x}+r^{t}_{y}\sigma _{y}+r^{t}_{z}\sigma _{z})$, with $r^{t}_{x}=r_{x}q_{t}$, $r^{t}_{y}=r_{y}q_{t}$, and $r^{t}_{z}=r_{z}$. According to the specific expressions of $\tau ^{M}_{qsl}$ and $\tau ^{D}_{qsl}$, it is easy to verify that $\tau ^{M}_{qsl}/\tau \equiv 1$, whether the initial state is pure or mixed. That is to say, by considering the dephasing channel, the bound derived in [21, 22] is the attainable tightest. However, by considering $\tau ^{U}_{qsl}$ for the mixed initial state and $\tau ^{D}_{qsl}$ for the pure initial state, we can acquire $\tau ^{U/D}_{qsl}/\tau =\sqrt{r^{2}_{x}+r^{2}_{y}}\le 1$. $\Vert r\Vert \le 1$ is the length of the GBV $\mathbf {r}$ for the initial state $\rho _{0}$. In the case of the pure initial state $\rho _{0}$, $\Vert r\Vert =1$ must be satisfied. When $r_{z}=0$, $\tau ^{M}_{qsl}/\tau =\tau ^{D}_{qsl}/\tau =1$, while for the mixed initial state, it is clear to find that $\tau ^{U}_{qsl}$ is always less than the actual evolutional time $\tau$. So in this case for the nonunitary dephasing channel, $\tau ^{M}_{qsl}/\tau >\tau ^{U}_{qsl}/\tau$ can be acquired.

According to the comparison of the above bounds in amplitude damping channel and the pure dephasing channel, the bound $\tau ^{M}_{qsl}$ is not only easier to compute and measure than Deffner’s bound, but also tighter than Deffner’s bound. So in the paper, we mainly use $\tau ^{M}_{qsl}$ as the proper QSLT to depict the maximal quantum evolution speed of the open system and explore the manipulation of quantum dynamical speedup of an open system.