Exploring the Limits of Controlled Markovian Quantum Dynamics with Thermal Resources

Abstract

Our aim is twofold: First, we rigorously analyse the generators of quantum-dynamical semigroups of thermodynamic processes. We characterise a wide class of GKSL-generators for quantum maps within thermal operations and argue that every infinitesimal generator of (a one-parameter semigroup of) Markovian thermal operations belongs to this class. We completely classify and visualise them and their non-Markovian counterparts for the case of a single qubit. Second, we use this description in the framework of bilinear control systems to characterise reachable sets of coherently controllable quantum systems with switchable coupling to a thermal bath. The core problem reduces to studying a hybrid control system ("toy model") on the standard simplex allowing for two types of evolution: (i) instantaneous permutations and (ii) a one-parameter semigroup of $d$-stochastic maps. We generalise upper bounds of the reachable set of this toy model invoking new results on thermomajorisation. Using tools of control theory we fully characterise these reachable sets as well as the set of stabilisable states as exemplified by exact results in qutrit systems.

Figures (7)

• Figure 1: (Colour online). Sketch of the semigroups $\mathsf{(En)TO}$ and $\mathsf{Gibbs}$, as well as their Markovian counterparts $\mathsf{M(En)TO}$ and $\mathsf{MGibbs}$ defined as Lie semigroups generated by their corresponding Lie wedges, see also the qubit example in Fig. \ref{['fig_TO_qubit_Markov']} . Note that Markovianity depends on the semigroup, i.e. there might be elements in $\mathsf{EnTO}\cap\mathsf{MGibbs}$ outside of $\mathsf{MEnTO}$. A similar set-inclusion for elementary thermal operations ($\mathsf{ETOs}$, which in general are a proper subset of $\overline{{\sf TO}}$s) can be found in Lostaglio18.
• Figure 2: (Colour online). Upper panels: Two aspect angles for the graphs of $\Psi_T$ (with $\varepsilon=0.6$) when restricting the domain to $\overline{\mathsf{TO}(H_0,T)}$ (blue cone), and its Markovian counterpart $\mathsf{MTO}(H_0,T)$ (orange cone), respectively. The yellow "tip" of the blue (orange) cone corresponds to the respective $\beta$-swaps (the "thermal reset" map $\rho\mapsto\rho_{\sf{Gibbs}}(H_0,T)$). --- Lower panel: The time dependence in $\mu_t$ formally leads to a time in multiples of the scaling factor $u > 0$ of $t[u]$, which is real in the Markovian segment $0\leq\mu_t\leq \mu_*$, i.e. up to the pole at $\mu_*$ and complex in the non-Markovian segment $\mu_*<\mu_t\leq 1$ as detailed in the text. The critical $\mu_*\in[\frac{1}{2},1]$ tends to one as $T\to 0^+$, thus illustrating $\mathsf{MTO}(H_0,T) \to \overline{\mathsf{TO}(H_0,T)}$ in the zero-temperature limit.
• Figure 3: (Colour online). Upper left: Evolutions of initial $x_0=(0.9, 0.07, 0.03)^\top$ and permutations $\pi(x_0)$ under $\mathbf{\Gamma}_d$ with $V_1,V_2$, $\theta=\tfrac{\pi}{6}$ of Eqs. \ref{['eq:sigma+']}-\ref{['eq:thermal_angle']} drive to fixed point $d$; upper right panel includes all permutations of trajectories starting with permutations of $d$, i.e. $x_0=\pi(d)$; the red region shows states $d$-majorised by $x_0$, blue regions are their permutations; the convex hull over red and blue regions contains entire reachable set $\mathfrak{reach}_{\Lambda_d}(x_0)$; inset gives the vector field to the dissipative part of the dynamics. Lower left: For $\theta=\tfrac{\pi}{5}$ in \ref{['eq:thermal_angle']}, as generically, the extreme point $z=\left(0.65, 0.30, 0.05\right)^\top$ in red differs from $x_0=\left(0.55, 0.40, 0.05\right)^\top$ as well as from $d=\left(0.55, 0.29, 0.16\right)^\top$. The lower right shows the vector fields under the full dynamics $\Lambda_d$ of dissipation and permutation control with stabilisable points (cp. Sec. \ref{['sec:Qutrit-Results']}, Fig. \ref{['fig:reach-stab']}) in turkish blue.
• Figure 4: (Colour online). Illustration of how to construct the boundary curves of the set of stabilisable points in the case of $a=0.3$. Left: For $\alpha=0-0.4-0.6$ the shaded regions comprise the points where the functionals $\alpha_\pi$ are negative; highlighted is the intersection of all the negative regions with the simplex. The points in this region are certainly not stabilisable. In particular the intersection point of $\ker(\alpha_{\mathrm{id}})$ and $\ker(\alpha_{\tau_{23}})$ is marked in red. Right: For three different values of $\alpha$, parts of $\ker(\alpha_{\mathrm{id}})$ and $\ker(\alpha_{\tau_{23}})$ and their intersections are shown. Taken together, these intersections form the curve given in red, which constitutes a part of the boundary of the set of stabilisable points.
• Figure 5: (Colour online). Left: The set of stabilisable states for the equidistant energy case with $a=\frac{1}{5}$. The set is bounded by six conics and contains the permutations of $d$. This is the elliptic case, and part of the ellipse is drawn in red, together with its permuted copies. The blue curve is obtained analogously by taking the hyperbolic case $a=5$ whose fixed point we denote $d'$. Right: The same approach gives the boundary of the set of stabilisable states for a random generator $B$ numerically. NB: In general the bounding curves need not be conic sections, and one may obtain a convex shape.

Theorems & Definitions (55)

• Proposition 1
• proof
• Proposition 2
• Corollary 1
• proof
• Theorem 1
• Remark 1
• Proposition 3
• Corollary 2
• Remark 2