Provably Time-Optimal Cooling of Markovian Quantum Systems

Abstract

We address the problem of cooling a Markovian quantum system to a pure state in the shortest amount of time possible. Here the system drift takes the form of a Lindblad master equation and we assume fast unitary control. This setting allows for a natural reduction of the control system to the eigenvalues of the state density matrix. We give a simple necessary and sufficient characterization of systems which are (asymptotically) coolable and present a powerful result which allows to considerably simplify the search for optimal cooling solutions. With these tools at our disposal we derive explicit provably time-optimal cooling protocols for rank one qubit systems, inverted $Λ$-systems on a qutrit, and a certain system consisting of two coupled qubits.

Figures (6)

• Figure 1: The parametrized space of generators $\mathfrak{Q}$ of a rank one system as given in Lemma \ref{['lemma:rank-one-space-of-lines']} and Corollary \ref{['coro:rank-one-space-of-lines']}. We work in a basis where $[V,V^*]$ is diagonal. The poles are mapped to the corners $(\pm(1-\nu^2),1+\nu^2)$. The latitude lines are vertical, and the equator lies on the $y$-axis. The longitude lines are parabolas passing through the poles and intersecting the $y$-axis between $b=\frac{1}{2}(1\pm\nu)^2$.
• Figure 2: Achievable derivatives as a set-valued function of $\lambda$ with the upper bound $\mu$, given in Lemma \ref{['lemma:mu']}, highlighted.
• Figure 3: Time-optimal path from the boundary of the Bloch ball (pure state) to the center (maximally mixed state) and back. Starting at the south pole, the path follows the $z$-axis until $\lambda=\lambda_0$. Then the path takes a sharp turn and continues horizontally until it reaches the boundary (which happens only asymptotically). When projected onto the $x,y$-plane, the horizontal part is a straight line lying on the negative $x$-axis (the mirrored path along the positive $x$-axis is also optimal). The black dots are equally spaced in time, and accumulate towards the end. The solution shares some similarity with the so-called magic plane result for the Bloch equations obtained in Lapert10.
• Figure 4: Optimal control function $u_y$ (solid) with direct (dashed) and compensation (dotted) contributions using $\nu=1/2$. The control is identically zero on $[0,t_0]$ and has a singularity at $t_0$.
• Figure 5: Achievable derivatives in the $\mathrm V$-system with $\gamma_1=1$ and $\gamma_2=2$ at the point $\lambda=(0.4,0.35,0.25)$.

Theorems & Definitions (26)

• Remark III.1
• Proposition III.2
• Theorem IV.1
• Lemma V.1
• proof
• Lemma V.2
• proof
• Corollary V.3
• Remark V.4
• Lemma V.5