Beating the limits with initial correlations

Assuming that a quantum system can be influenced by external fields $\{{\varepsilon }_{k}(t)\}$, OCT provides the means to maximize or minimize a predefined figure of merit. In our case, the control problem is a simple state-to-state transfer [32], achieved within a fixed time T. The total optimization functional,

$$\begin{eqnarray}&&F[\{{\varepsilon }_{k}\}]={\epsilon }_{T}[\hat{{\boldsymbol{\rho }}}(T)]+{\int }_{0}^{T}{\rm{d}}t\,g[\{{\varepsilon }_{k}(t)\},\hat{{\boldsymbol{\rho }}}(t),t],\end{eqnarray} \tag{ 9 }$$

consists of the figure of merit ${\epsilon }_{T}[\hat{{\boldsymbol{\rho }}}(T)]$ and additional constraints, captured in a function g. In the following, we consider only a single external field, $\varepsilon (t)$. Our figure of merit is the error in preparing the qubit in the desired target state, irrespective of the TLS state. This can be expressed as [33]

$$\begin{eqnarray}&&{\epsilon }_{T}[\hat{{\boldsymbol{\rho }}}(T)]=1-\phantom{\rule{}{2.0ex}}\langle {{\rm{\Psi }}}_{{\rm{Q}}}^{\mathrm{targ}}\phantom{\rule{}{2.0ex}}| {{\rm{Tr}}}_{\mathrm{TLS}}^{}[\hat{{\boldsymbol{\rho }}}(T)]\phantom{\rule{}{2.0ex}}| {{\rm{\Psi }}}_{{\rm{Q}}}^{\mathrm{targ}}\phantom{\rule{}{2.0ex}}\rangle ,\end{eqnarray} \tag{ 10 }$$

where ${{\rm{Tr}}}_{\mathrm{TLS}}^{}[\cdot ]$ describes the partial trace over the TLS. Without loss of generality, we choose the target state $| {{\rm{\Psi }}}_{{\rm{Q}}}^{\mathrm{targ}}\rangle$ to be the bare ground state of the qubit.

We will use Krotov's method [34], an iterative optimization algorithm with built-in monotonic convergence [35], in the following. The constraint function is chosen as

$$\begin{eqnarray}&&g[\{\varepsilon (t)\}]=\displaystyle \frac{\lambda }{S(t)}{(\varepsilon (t)-{\varepsilon }^{\mathrm{ref}}(t))}^{2},\end{eqnarray} \tag{ 11 }$$

where λ is a numerical parameter that controls the update magnitude of the field $\varepsilon (t)$, S(t) a shape function and ${\varepsilon }^{\mathrm{ref}}(t)$ a reference field (taken to be the field from the previous iteration). The actual update equation from the fields is determined by equation (11), the equation of motion (2) and the final time target (10). For more details see [35].

We start by deriving the optimal reset protocol for a factorizing initial state (5) of qubit and TLS, i.e., when no initial correlation between system and environment, i.e., between qubit and TLS, is present. Note that the level splittings of qubit and TLS are not the same and ${\omega }_{\mathrm{TLS}}\gt {\omega }_{{\rm{Q}}}$. This, together with the identical temperature of qubit and TLS, results in a higher von Neumann entropy of qubit than TLS. According to the second law of thermodynamics, one would expect the best cooling to be achieved by an entropy exchange between TLS and qubit. This has indeed been observed before [14, 15].

For the chosen parameters, entropy exchange can be realized by simply swapping the ground state populations of qubit and TLS. This is best achieved when qubit and TLS are in resonance. As can be seen in equation (1), the control field $\varepsilon (t)$ effectively changes the level splitting of the qubit. Therefore an educated guess would be to ramp qubit and TLS rapidly into resonance and stay there just long enough for a full swap operation. Figure 2(a) shows the dynamics for this particular guess field (dashed lines), as well as the free evolution (dotted lines) and the dynamics under the optimized field (solid lines). With the optimized field, we indeed obtain the anticipated swap in the ground state populations at t = T. In contrast, for the guess field, the maximal ${p}_{{\rm{Q}}}(t)$ is already achieved at $t\approx 17$.

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As we will show analytically in section 4 below, the swap is the best and fastest protocol for all factorizing initial conditions when the TLS is initially diagonal in its eigenbasis. The analytical bounds for the minimal error and the shortest possible duration in which the minimal error is reached, given the parameters used in figure 2, are

$$\begin{eqnarray}&&{\epsilon }_{T}^{\min }=1-{p}_{\mathrm{TLS}}^{\mathrm{th}}=4.74 \% ,\quad {T}^{\min }=\displaystyle \frac{\pi }{2J}=15.7.\end{eqnarray} \tag{ 12 }$$

The actual value of the minimal error ${\epsilon }_{T}^{\min }$ is determined by the initial ground state population ${p}_{\mathrm{TLS}}^{\mathrm{th}}$ of the TLS, i.e., it is governed by the reservoir temperature. One might wonder why the minimal time $t\approx 17$ required for the swap in figure 2(a) is larger than ${T}^{\min }$ in equation (12). This is due to the fact that, for the sake of experimentally feasible control signals, we do not allow $\varepsilon (t)$ to be instantaneously switched on and off. If we relax this constraint, our optimized control reaches the quantum speed limit ${T}^{\min }$.

For any time longer than ${T}^{\min }$, there is always at least one solution achieving maximal cooling but there may be more, i.e., the control strategy is not unique. Another possible control field is shown exemplarily in figure 2(d). The non-uniqueness of the solution allows for taking into account further experimentally desirable features, such as restriction of the maximal amplitude of the control, without losing performance.

One may wonder how robust these solutions are to noise in the controls or in the initial state. We have quantified the robustness of the dynamics shown in figure 2(a) by averaging over 1000 realizations of Gaussian amplitude noise for the optimized field shown in figure 2(b). For a typical noise level of 1% in the control amplitude, added in form of a varying scaling factor to the control, the final error increases by only a small amount, from 5.04% to 5.16% on average. In order to simulate noise in the initial state, equation (5), we have added Gaussian noise to the input parameters ${\omega }_{{\rm{Q}}}$, ${\omega }_{\mathrm{TLS}}$ and β, using again 1000 realizations. For noise levels up to 2%, we obtain no change in the error at all, and even 10% of state noise increase the error only from 5.04% to 5.18% on average. The protocol is thus very robust with respect to noise in the initial state. The reason for this finding will become clear below in section 4.

An obvious choice for a correlated initial state is the joint thermal equilibrium state (6) of qubit and TLS. For the chosen parameters, the mutual information of this state is rather small, ${{ \mathcal I }}^{\mathrm{init}}=4.0\times {10}^{-3}$. The state is separable but has non-zero quantum discord. Note that all initial states studied within this section have non-vanishing quantum discord, since for a thermalized TLS there is no state with only classical correlations.

As can be seen in figures 3(a)–(c), both cooling and erasure of correlation is achieved by the optimized control field. The final value of the error in figure 3(a), ${\epsilon }_{T}=4.74 \%$, coincides with the minimal error ${\epsilon }_{T}^{\min }$ for factorizing initial states, see equation (12). Optimal control therefore allows us to erase initial correlations. A robustness analysis analogous to that for figure 2 yields very similar results: amplitude noise at a level of 1% increases the error from 4.74% to 4.97%, whereas noise in the state has no effect at all up to the 2% level. It increases the error to only 4.85% at the 10% level.

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To further investigate the role of initial correlations, we now choose qubit and memory TLS to be in resonance, i.e. ${\omega }_{{\rm{Q}}}={\omega }_{\mathrm{TLS}}$. For factorizing initial conditions, no cooling at all would be possible. Additionally, we enhance the correlations, the initial state is given by equation (8). It is thermal in the sense that, if TLS or qubit is traced out, one obtains equation (4). Surprisingly, we are not only able to erase the correlations, but even achieve further cooling of the system, as can be seen in figures 3(d)–(f), for an initial state with mutual information ${{ \mathcal I }}^{\mathrm{init}}=0.345$ and quantum discord ${{ \mathcal Q }}^{\mathrm{init}}=0.228$. This is clear evidence for system-environment correlations acting as a resource for cooling.

Remarkably, even the speed limit obtained for factorizing initial conditions does not hold anymore. As can be seen in figure 4(a), with increasing total correlations, i.e., mutual information, the error threshold of the factorizing dynamics, ${\epsilon }_{T}^{\min }$, can be reached in shorter times. Note that although the upper left point in figure 4(a) lies above the approximate quantum speed limit for factorizing initial conditions, this is only due to influence of the counter rotating terms (which we will analyze in more detail in appendix A). If we temporarily neglect the counter rotating terms, the result coincides with the quantum speed limit.

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Moreover, figure 4(b) shows that the final error ${\epsilon }_{T}$ is reduced for increasing initial correlations. While we have also studied entangled initial states, the data is not presented here, as the results do not differ. We find that only the amount of mutual information, i.e., the total amount of correlation, not the type, i.e., classical or quantum correlations, is relevant for cooling.

A natural question is whether the speed limit reported in figure 4 depends on the type of control over the qubit. It turns out that a control field that couples to the system via ${\hat{{\boldsymbol{\sigma }}}}_{{\rm{Q}}}^{x}$ instead of ${\hat{{\boldsymbol{\sigma }}}}_{{\rm{Q}}}^{z}$ in equation (1) does not perform better (data not shown). We have found solutions swapping the populations between qubit and memory TLS also for that type of control when starting from factorizing initial states. Similarly to ${\hat{{\boldsymbol{\sigma }}}}_{{\rm{Q}}}^{z}$-control, correlations in the initial state allow for better reset with smaller errors. However, more time is required in both cases when the control couples via ${\hat{{\boldsymbol{\sigma }}}}_{{\rm{Q}}}^{x}$. As a consequence, the weakly coupled reservoir has a larger impact on the dynamics.

To summarize our findings obtained so far, it is not only possible to reset the qubit in the presence of initial correlations; initial correlations between system and environment can actually be used to enhance the performance of the cooling protocol. Moreover, in the resonant case, initial correlations enable cooling that is impossible without their presence. We analyze the dynamics that lead to this surprising result in more detail in section 4.

Finally, we investigate whether non-Markovianity of the dynamics has any influence on the optimized fields and achievable final errors. The dynamics of the qubit becomes Markovian or non-Markovian depending on the ratio $J/\kappa$. We quantify this by the accessible volume of state space [36] to study a possible interplay between non-Markovianity and control.

In our setup, we observe that non-Markovianity seems to be linked to population flow between qubit and memory TLS. More precisely, a monotonic decrease in the qubit's state space volume can be observed, when populations flows from the memory TLS into the qubit, i.e., increasing the ground state population of the qubit while decreasing it for the memory TLS. This hints towards Markovian dynamics. In contrast, an increase in the state space volume occurs for the reversed population flow, indicating non-Markovian dynamics.

The population flow between qubit and memory TLS is governed by their effective coupling. It is directly influenced by the coupling J and indirectly by the relative detuning $\delta (t)={\omega }_{{\rm{Q}}}+\varepsilon (t)-{\omega }_{\mathrm{TLS}}$ between both. The frequency, with which the population flow changes its direction, increases with $| \delta (t)|$, while its smallest value is assumed for $\delta (t)=0$, where the frequency is entirely determined by J. According to this observation, the dynamics of the time-optimal solution (see equation (12)) turns out to be Markovian. In this case, the ground state population of the qubit is constantly increasing until reaching its maximum at ${T}^{\min }$. For longer times and non-optimal driving, the controlled dynamics can become non-Markovian, see figures 2(c) and (d), as the population flows in both directions at intermediate times. Nevertheless, implementing a swap at $T\gt {T}^{\min }$ is also possible with entirely Markovian dynamics, see figures 2(a) and (b). This shows that even though non-Markovianity is not crucial for the qubit reset, it also is not harmful in the sense that the optimization does not suppress non-Markovianity.

In the following we use concepts from geometric control theory [37], where the idea consists in transforming the dynamical equations of the system in such a way that the optimality condition can be expressed analytically [38, 39]. For ease of the derivation, we transform states and Hamiltonian into the rotating frame. Neglecting the counter-rotating terms and the (slow) equilibration with the reservoir, the equation of motion reads

$$\begin{eqnarray}&&{\rm{i}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}\hat{{\boldsymbol{\rho }}}^{\prime} (t)=\phantom{\rule{}{2.0ex}}[\hat{{\boldsymbol\sf{H}}}^{\prime} (t),\hat{{\boldsymbol{\rho }}}^{\prime} (t)\phantom{\rule{}{2.0ex}}],\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}\hat{{\boldsymbol\sf{H}}}^{\prime} (t)=\left(\begin{array}{cccc}\tfrac{{\rm{d}}\delta (t)}{{\rm{d}}t}\tfrac{t}{2} & 0 & 0 & 0\\ 0 & \tfrac{{\rm{d}}\delta (t)}{{\rm{d}}t}\tfrac{t}{2} & J(t){{\rm{e}}}^{-{\rm{i}}\delta (t)t} & 0\\ 0 & J(t){{\rm{e}}}^{{\rm{i}}\delta (t)t} & -\tfrac{{\rm{d}}\delta (t)}{{\rm{d}}t}\tfrac{t}{2} & 0\\ 0 & 0 & 0 & -\tfrac{{\rm{d}}\delta (t)}{{\rm{d}}t}\tfrac{t}{2}\end{array}\right),\end{eqnarray} \tag{ 13b }$$

where $\delta (t)={\omega }_{{\rm{Q}}}+\varepsilon (t)-{\omega }_{\mathrm{TLS}}$ is the time-dependent detuning of qubit and TLS. For the sake of generality, we account for a possible time-dependence $J=J(t)$ of the coupling strength between qubit and TLS.

For the numerical optimization in section 3, the optimization target was to reset the qubit to its ground state. Here, we choose a more general approach and maximize the qubit's purity⁶ . The key idea in the following is to chose a representation of the state $\hat{{\boldsymbol{\rho }}}^{\prime} (t)$ in terms of a set of real variables $\{{x}_{1}(t),\,...,\,{x}_{16}(t)\}$ to span the entire state space of qubit and TLS. Inserting this representation into equation (13a), one obtains coupled equations for all x_i. In order to decouple these equations and reduce the number of relevant variables, one needs to perform an appropriate variable transformation $\{{x}_{1}(t),\,...,\,{x}_{16}(t)\}\to \{{z}_{1}(t),\,...,\,{z}_{16}(t)\}$. A more detailed description of the transformations can be found in appendix B.

In the new variables, the qubit's purity becomes

$$\begin{eqnarray}&&{{ \mathcal P }}_{{\rm{Q}}}=\displaystyle \frac{1}{2}+2\phantom{\rule{}{2.0ex}}({z}_{1}^{2}+{z}_{5}^{2}+{z}_{7}^{2}\phantom{\rule{}{2.0ex}}),\end{eqnarray} \tag{ 14 }$$

where we have dropped the explicit time dependence for all quantities. The corresponding equations of motion are decoupled into two separate subspaces. On the one hand, we have

$$\begin{eqnarray}&&\left(\begin{array}{c}{\dot{z}}_{1}\\ {\dot{z}}_{2}\\ {\dot{z}}_{3}\end{array}\right)=2{J}_{1}\left(\begin{array}{c}-{z}_{2}\\ {z}_{1}-{z}_{1}^{{\rm{c}}}\\ 0\end{array}\right)+2{J}_{2}\left(\begin{array}{c}-{z}_{3}\\ 0\\ {z}_{1}-{z}_{1}^{{\rm{c}}}\end{array}\right)+2\alpha \left(\begin{array}{c}0\\ -{z}_{3}\\ {z}_{2}\end{array}\right),\end{eqnarray} \tag{ 15 }$$

describing the dynamics of the qubit's ground state population, ${p}_{{\rm{Q}}}={z}_{1}+1/2$, within the three-dimensional subspace ${S}_{1}=\{{z}_{1},{z}_{2},{z}_{3}\}$, ${z}_{1}^{{\rm{c}}}$ being a constant. Note that z₂, z₃ are non-zero at time t = 0 only if initial correlations are present, see equations (8), (B4) and (B7). Equation (15) thus already indicates that initial correlations can be transferred into ground state population and hence purity. On the other hand, the qubit's coherences, ${\gamma }_{{\rm{Q}}}={z}_{5}+{\rm{i}}{z}_{7}$, evolve within the four-dimensional subspace ${S}_{2}=\{{z}_{5},{z}_{6},{z}_{7},{z}_{8}\}$,

$$\begin{eqnarray}&&\left(\begin{array}{c}{\dot{z}}_{5}\\ {\dot{z}}_{6}\\ {\dot{z}}_{7}\\ {\dot{z}}_{8}\end{array}\right)={J}_{1}\left(\begin{array}{c}-{z}_{6}\\ {z}_{5}\\ {z}_{8}\\ -{z}_{7}\end{array}\right)+{J}_{2}\left(\begin{array}{c}{z}_{8}\\ -{z}_{7}\\ {z}_{6}\\ -{z}_{5}\end{array}\right)+2\alpha \left(\begin{array}{c}{z}_{7}\\ 0\\ -{z}_{5}\\ 0\end{array}\right),\end{eqnarray} \tag{ 16 }$$

where z₆ and z₈ are related, but not equivalent, to the TLS coherence. The three fields are given by

$$\begin{eqnarray}&&{J}_{1}=J\cos (\delta t),\quad {J}_{2}=J\sin (\delta t),\quad \alpha =\displaystyle \frac{1}{2}\displaystyle \frac{{\rm{d}}\delta }{{\rm{d}}t}t.\end{eqnarray} \tag{ 17 }$$

It is straightforward to show that the dynamics within the subspaces S₁ and S₂ is restricted to the surface of two spheres. For S₁, we find from equation (15)

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{R}_{1}^{2}=0,\qquad {R}_{1}=\sqrt{{({z}_{1}-{z}_{1}^{{\rm{c}}})}^{2}+{z}_{2}^{2}+{z}_{3}^{2}},\end{eqnarray} \tag{ 18 }$$

with R₁ the radius of the sphere centered around $({z}_{1}^{{\rm{c}}},0,0)$ with constant ${z}_{1}^{{\rm{c}}}=-({z}_{4}+1)/2$, see equation (B7). Similarly for S₂, equation (16) yields

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{R}_{2}^{2}=0,\qquad {R}_{2}=\sqrt{{z}_{5}^{2}+{z}_{6}^{2}+{z}_{7}^{2}+{z}_{8}^{2}},\end{eqnarray} \tag{ 19 }$$

with radius R₂ and center $(0,0,0,0)$. The values of R₁ and R₂ are determined by the initial values ${z}_{i}^{\mathrm{init}}$ with $i=1,\,\ldots ,\,8$. In other words, the accessible part of the entire state space is fully determined by the initial state ${\hat{{\boldsymbol{\rho }}}}^{\mathrm{init}}=\hat{{\boldsymbol{\rho }}}(0)=\hat{{\boldsymbol{\rho }}}^{\prime} (0)$.

The factorizing initial state (5) is obviously diagonal. Thus we have ${z}_{2}^{\mathrm{init}}={z}_{3}^{\mathrm{init}}=0$ as well as ${z}_{i}^{\mathrm{init}}=0$, $i=5,\,...,\,8$. As a consequence, ${R}_{2}=0$, i.e., no dynamics will occur in S₂, and the relevant subspace is entirely given by S₁. In the following, we parametrize equation (5) as

$$\begin{eqnarray}{\hat{{\boldsymbol{\rho }}}}^{\mathrm{init}}={\hat{{\boldsymbol{\rho }}}}_{{\rm{Q}}}^{\mathrm{th}}\otimes {\hat{{\boldsymbol{\rho }}}}_{\mathrm{TLS}}^{\mathrm{th}}=\left(\begin{array}{cc}{a}_{{\rm{Q}}} & 0\\ 0 & {b}_{{\rm{Q}}}\end{array}\right)\otimes \left(\begin{array}{cc}{a}_{\mathrm{TLS}} & 0\\ 0 & {b}_{\mathrm{TLS}}\end{array}\right),\end{eqnarray} \tag{ 20 }$$

and assume ${\hat{{\boldsymbol{\rho }}}}_{\mathrm{TLS}}^{\mathrm{th}}$ to be initially more pure than ${\hat{{\boldsymbol{\rho }}}}_{{\rm{Q}}}^{\mathrm{th}}$. This amounts to ${a}_{\mathrm{TLS}}^{2}+{b}_{\mathrm{TLS}}^{2}\gt {a}_{{\rm{Q}}}^{2}+{b}_{{\rm{Q}}}^{2}$ with $a,b$ the ground and excited state populations of qubit and TLS, respectively. We first discuss the resonant case, i.e., $\delta =0$ for all t, and derive the time-optimal solution for the control problem. Second, we show that allowing for $\delta \ne 0$ does not improve the best possible final purity of the qubit.

For $\delta =0$ for all t, which implies ${J}_{1}=J$ and ${J}_{2}=\alpha =0$, equation (15) is further simplified and the dynamics are confined to the two-dimensional subspace ${S}_{1}^{2}=\{{z}_{1},{z}_{2}\}$,

$$\begin{eqnarray}&&\left(\begin{array}{c}{\dot{z}}_{1}\\ {\dot{z}}_{2}\end{array}\right)=2J\left(\begin{array}{c}-{z}_{2}\\ {z}_{1}-{z}_{1}^{{\rm{c}}}\end{array}\right).\end{eqnarray} \tag{ 21 }$$

Figure 5 shows the accessible state space for the dynamics within ${S}_{1}^{2}$ when starting in the initial state used in figure 2. Depending on the sign of J, the initial state evolves along the vector field ($J\gt 0$) or opposite to it ($J\lt 0$), see equation (21). The optimization target can then be trivially identified as the point with maximal z₁ on this curve. Assuming constant positive coupling J, the state will evolve with constant speed along the green line in figure 5. It then takes ${T}^{\min }=\pi /(2J)$ to reach the rightmost point. This can simply be shown by integrating along the green line. Allowing for time-dependent coupling $J(t)\geqslant 0$, the minimal time is given by

$$\begin{eqnarray}&&{\int }_{0}^{{T}^{\min }}J(t){\rm{d}}t=\displaystyle \frac{\pi }{2}.\end{eqnarray} \tag{ 22 }$$

Therefore, the time-optimal solution is to choose J(t) maximal for all t.

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The point of maximum qubit purity, ${{ \mathcal P }}_{{\rm{Q}}}^{\max }$, is determined by the center ${z}_{1}^{{\rm{c}}}$ of the sphere and its radius R₁,

$$\begin{eqnarray}&&{{ \mathcal P }}_{{\rm{Q}}}^{\max }=\displaystyle \frac{1}{2}+2{({z}_{1}^{{\rm{c}}}+{R}_{1})}^{2}={a}_{\mathrm{TLS}}^{2}+{b}_{\mathrm{TLS}}^{2}={{ \mathcal P }}_{\mathrm{TLS}}^{\mathrm{init}},\end{eqnarray} \tag{ 23 }$$

with ${{ \mathcal P }}_{\mathrm{TLS}}^{\mathrm{init}}$ the initial TLS purity. Equations (22) and (23) hold for any initial factorizing state of the form (20) with the TLS initially purer than the qubit. Note that for ${z}_{1}^{{\rm{c}}}\lt 0$, equation (23) becomes ${{ \mathcal P }}_{{\rm{Q}}}^{\max }=\tfrac{1}{2}+2{({z}_{1}^{{\rm{c}}}-{R}_{1})}^{2}$ but yields identical results.

It is straightforward to see that a non-vanishing time-dependent detuning $\delta \ne 0$ does not provide access to states with higher qubit purity. The dynamics is confined to the surface of the three-dimensional sphere S₁, see equation (18), and the point of maximal purity is already accessible with $\delta =0$ for all t. It is important to note that $\delta \ne 0$ involves dynamics in the z₃-dimension. This becomes crucial when starting with initially correlated states.

The most general initially factorizing state for qubit and TLS is given by

$$\begin{eqnarray}{\hat{{\boldsymbol{\rho }}}}^{\mathrm{init}}={\hat{{\boldsymbol{\rho }}}}_{{\rm{Q}}}\otimes {\hat{{\boldsymbol{\rho }}}}_{\mathrm{TLS}}=\left(\begin{array}{cc}{a}_{{\rm{Q}}} & {\gamma }_{{\rm{Q}}}\\ {\gamma }_{{\rm{Q}}}^{* } & {b}_{{\rm{Q}}}\end{array}\right)\otimes \left(\begin{array}{cc}{a}_{\mathrm{TLS}} & {\gamma }_{\mathrm{TLS}}\\ {\gamma }_{\mathrm{TLS}}^{* } & {b}_{\mathrm{TLS}}\end{array}\right),\end{eqnarray} \tag{ 24 }$$

with $a,b$ as in equation (20) and ${\gamma }_{{\rm{Q}}},{\gamma }_{\mathrm{TLS}}$ the coherences of qubit and TLS. We first consider the case ${\gamma }_{\mathrm{TLS}}=0$. From a physical perspective, this is a well justified initial state, since we assume the TLS to be in permanent contact with the reservoir and thus in thermal equilibrium. In contrast, for the qubit, non-zero coherences, ${\gamma }_{{\rm{Q}}}\ne 0$, are a possible scenario, e.g., as a result of its previous use in a computation. In this case, we again find ${z}_{2}^{\mathrm{init}}={z}_{3}^{\mathrm{init}}=0$. However, ${z}_{5}={\mathfrak{Re}}\{{\gamma }_{{\rm{Q}}}\}$ or ${z}_{7}={\mathfrak{Im}}\{{\gamma }_{{\rm{Q}}}\}$ or both will be non-zero. Note that ${z}_{6}^{\mathrm{init}}={z}_{8}^{\mathrm{init}}=0$ still holds but there is dynamics within the subspace S₂, since ${R}_{2}\ne 0$.

Assuming resonance in the following (i.e., $\delta =0$ for all t), the dynamics within S₁ is reduced to the two-dimensional subspace ${S}_{1}^{2}$, as discussed before. Similarly, the dynamics in the four-dimensional subspace S₂ decouple and can be described by two two-dimensional subspaces, ${S}_{2}^{2}$ and ${S}_{3}^{2}$. Their respective equations of motions are

$$\begin{eqnarray}&&\left(\begin{array}{c}{\dot{z}}_{5}\\ {\dot{z}}_{6}\end{array}\right)=J\left(\begin{array}{c}-{z}_{6}\\ {z}_{5}\end{array}\right),\qquad \left(\begin{array}{c}{\dot{z}}_{7}\\ {\dot{z}}_{8}\end{array}\right)=J\left(\begin{array}{c}{z}_{8}\\ -{z}_{7}\end{array}\right).\end{eqnarray} \tag{ 25 }$$

Figure 6 shows the evolution in the three subspaces ${S}_{1}^{2}$, ${S}_{2}^{2}$ and ${S}_{3}^{2}$ for an exemplary initial factorizing state with ${\gamma }_{{\rm{Q}}}\ne 0$ and ${\gamma }_{\mathrm{TLS}}=0$. We now have dynamics in all three subspaces. As before, maximizing the qubit's ground state population, ${p}_{{\rm{Q}}}={z}_{1}+1/2$, requires the time $T=\pi /(2J)$, which corresponds to evolution in terms of a half circle in ${S}_{1}^{2}$. Importantly, the motion within ${S}_{1}^{2}$ is twice as fast as that in ${S}_{2}^{2}$ and ${S}_{3}^{2}$, which can be easily seen by comparing equations (21) and (25). Therefore, at time $T=\pi /(2J)$, the qubit's coherences, ${\gamma }_{{\rm{Q}}}={z}_{5}+{\rm{i}}{z}_{7}$, vanish, since the evolution within ${S}_{2}^{2}$ and ${S}_{3}^{2}$ only runs through a quarter circle. The minimal reset time is thus not changed when allowing for coherences in the initial qubit state. This finding is in line with the observation that for pure states (as considered in this section), standard quantum speed limit bounds coincide with the bound obtained from the Wigner–Yanase skew information which particularly quantifies the coherence of a state (relative to the eigenbasis of the Hamiltonian)⁷ [40, 41]. Moreover, as long as the initial purities of qubit and TLS satisfy ${{ \mathcal P }}_{{\rm{Q}}}^{\mathrm{init}}\lt {{ \mathcal P }}_{\mathrm{TLS}}^{\mathrm{init}}$, the time-optimal solution is still the swap operation given by equations (22) and (23). This is true irrespective of the specific initial state of the qubit.

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If we allow for coherences also in the initial state of the TLS, ${\gamma }_{\mathrm{TLS}}\ne 0$, this does not hold anymore. In this case, some or all of the initial values ${z}_{2}^{\mathrm{init}}$, ${z}_{3}^{\mathrm{init}}$, ${z}_{6}^{\mathrm{init}}$ and ${z}_{8}^{\mathrm{init}}$ are non-zero. Geometrically, the large dots in the three spheres ${S}_{1}^{2}$, ${S}_{2}^{2}$ and ${S}_{3}^{2}$ in figure 6 are then placed at arbitrary points along the green curves. Thus, the evolution cannot easily be synchronized in terms of half and quarter circles. Rather, exact knowledge of the initial state would be required to determine the optimal solution.

For correlated initial states, the dynamics involving the qubit ground state population z₁ explores all three dimensions of the subspace S₁ spanned by ${z}_{1},{z}_{2},{z}_{3}$. We show that a geometric analysis is still useful in this case since it provides physical insight into the control mechanisms of the optimal solution. In particular, it explains why initial correlations result in a higher purity and a shorter time for the reset.

For any initial state satisfying equation (8), no dynamics occurs in ${S}_{2}$ . It is then straightforward to show that these correlated initial states allow to access states with higher purity than factorizing states: Since the reduced states of qubit and TLS are unchanged by the presence of correlations, the center $({z}_{1}^{{\rm{c}}},0,0)$ of the sphere in S₁ remains the same, while its radius R₁ increases, see equation (18). As a result, the set of accessible states that may be reached by the dynamics is enlarged.

Figure 7 shows the evolution starting from a correlated initial state under a field designed by numerical optimization. It illustrates why the quantum speed limit for factorizing initial states can be beaten. For the initial state in figure 7, ${z}_{2}^{\mathrm{init}}=0$ and ${z}_{3}^{\mathrm{init}}\lt 0$. The optimized field drives the state rapidly towards the ${z}_{3}=0$ plane. This is achieved by the characteristic off-resonant peak in the optimized field between t = 0 and t = 2. The subsequent evolution with $\delta =0$ becomes two-dimensional within the z₁–z₂ plane; it is equivalent to that in figure 5 discussed above. However, in contrast to the dynamics shown in figure 5, the motion in the z₁–z₂ plane has to overcome a reduced distance as a consequence of the initial transfer between ${z}_{3}\lt 0$ and ${z}_{3}=0$. It can be seen from the projection of the entire motion onto the z₁–z₂ plane (shown in the front left plane in figure 7 top, note in particular the position of the third small dot), that less than a half circle has to be overcome by the evolution with $\delta =0$ to reach the point of largest purity, ${z}_{1,\max }={z}_{1}^{{\rm{c}}}+{R}_{1}$. Since the initial transfer towards the ${z}_{3}=0$ plane is accomplished faster than any motion within this plane, the total time is reduced. Unfortunately, however, the reduction in time comes at a cost, namely the control field must be tuned to the initial value of z₃. In other words, for correlated initial states, derivation of the optimal control strategy requires knowledge of the initial state.

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This analysis can be completed by a geometric description of the solution. To this end, we consider the differential system (15) and assume the coupling J to be bounded, while there is no constraint on $\tfrac{{\rm{d}}\delta }{{\rm{d}}t}$, i.e., on $\alpha (t)$, see equation (17). As in the numerical optimization, the optimal solution can be decomposed into two steps. In a first stage, we neglect the first two terms on the right-hand side of equation (15) and the α-term is used to move arbitrarily fast in the z₂–z₃ plane from the initial point into the ${z}_{3}=0$ plane. This motion is completed in a short time ${\tau }_{\varepsilon }$ provided $\alpha (t)$ satisfies the condition

$$\begin{eqnarray}&&{\int }_{0}^{{\tau }_{\varepsilon }}2\alpha (t){\rm{d}}t=\displaystyle \frac{\pi }{2},\end{eqnarray} \tag{ 26 }$$

which, after integration by parts, leads to

$$\begin{eqnarray}&&\delta ({\tau }_{\varepsilon })-{\int }_{0}^{{\tau }_{\varepsilon }}\delta (t){\rm{d}}t=\displaystyle \frac{\pi }{2}.\end{eqnarray} \tag{ 27 }$$

A standard solution for δ is given by a linear time evolution of the form

$$\begin{eqnarray}&&\delta (t)=\displaystyle \frac{\pi t}{2{\tau }_{\varepsilon }(1-{\tau }_{\varepsilon }/2)}\quad \mathrm{for}\quad t\in [0,{\tau }_{\varepsilon }].\end{eqnarray} \tag{ 28 }$$

The second part of the optimal solution is the meridian trajectory in the ${z}_{3}=0$ plane with $\delta (t)=0$. In fact, for $\delta (t)$ constant, we recover the Grushin model [42]. It can be shown (using the appendix of [42]) that the meridian trajectory is the solution minimizing the time to reach the state of largest purity, ${z}_{1,{\rm{\max }}}$. The time required for the motion along the meridian is fixed by the initial point of this dynamics, it is ${T}^{{\rm{\min }}}={\theta }^{\mathrm{init}}/(2J)$ where ${\theta }^{\mathrm{init}}$ is the polar angle of the sphere S₁ given by ${z}_{1}^{\mathrm{init}}={R}_{1}\cos ({\theta }^{\mathrm{init}})$. Assuming the time to reach the ${z}_{3}=0$ plane, ${\tau }_{\varepsilon }$, to be arbitrarily small, the time ${\tau }_{\varepsilon }+{T}^{{\rm{\min }}}$ required for both steps of the time-optimal solution for correlated initial states is smaller than the time of $\pi /(2J)$ obtained with factorizing initial states. This rigorously confirms the role of initial correlations for the speedup of the purification process.

The robustness of the numerical control solutions with respect to noise in either control amplitude or initial state, observed in section 3, can be rationalized by the analytical solutions found here. Key to all of the reset strategies is a population swap between qubit and TLS. This is independent of the actual populations, as evidenced in the remarkable robustness with respect to noise in the initial state. The population swap requires resonance between qubit and TLS. Amplitude noise up to a level of 1% does not perturb the resonance sufficiently to have a noticeable effect on the final errors.