Searching for Lindbladians obeying local conservation laws and showing thermalization

Abstract

We investigate the possibility of a Markovian quantum master equation (QME) that consistently describes a finite-dimensional system, a part of which is weakly coupled to a thermal bath. In order to preserve complete positivity and trace, such a QME must be of Lindblad form. For physical consistency, it should additionally preserve local conservation laws and be able to show thermalization. We search of Lindblad equations satisfying these additional criteria. First, we show that the microscopically derived Bloch-Redfield equation (RE) violates complete positivity unless in extremely special cases. We then prove that imposing complete positivity and demanding preservation of local conservation laws enforces the Lindblad operators and the lamb-shift Hamiltonian to be `local', i.e, to be supported only on the part of the system directly coupled to the bath. We then cast the problem of finding `local' Lindblad QME which can show thermalization into a semidefinite program (SDP). We call this the thermalization optimization problem (TOP). For given system parameters and temperature, the solution of the TOP conclusively shows whether the desired type of QME is possible up to a given precision. Whenever possible, it also outputs a form for such a QME. For a XXZ chain of few qubits, fixing a reasonably high precision, we find that such a QME is impossible over a considerably wide parameter regime when only the first qubit is coupled to the bath. Remarkably, we find that when the first two qubits are attached to the bath, such a QME becomes possible over much of the same paramater regime, including a wide range of temperatures.

Figures (8)

• Figure 1: Schematic of the setup we consider: an arbitrary finite dimensional system described by Hamiltonian $H_S$, a part of which is weakly coupled to a thermal bath at inverse temperature $\beta$. The Hilbert space of the system, $\mathcal{H}_S$ is divided into a part $\mathcal{H}_L$ which directly couples to the bath, and the remaining part $\mathcal{H}_M$.
• Figure 2: $\tau_{opt}$ vs $g$, for $N_L = 1$, with $\omega^{(\ell)}_0 = 1$, $\Delta_\ell = 1$, $\beta = 1$ and $g_\ell = g$ for all $\ell$. The tolerance chosen is $\delta=10^{-6}$. We find that $\tau_{opt} \gg \delta$, conclusively showing that, for such setups, no QME can simultaneously preserve complete positivity, obey local conservation laws, and show thermalization up to the precision set by the tolerance.
• Figure 3: $\tau_{opt}$ vs $\beta$, for $N_L = 1$, with $\omega^{(\ell)}_0 = 1$, $\Delta_\ell = 1$, $g_\ell= 0.1$ for all $\ell$. The tolerance chosen is $\delta=10^{-6}$, and is plotted as the dashed horizontal line. We find that $\tau_{opt} \gg \delta$, indicating that, for such setups, it is not possible to have a QME simultaneously preserving complete positivity, obeying local conservation laws, and showing thermalization up to the precision set by the tolerance.
• Figure 4: $\tau_{opt}$ vs $g$, for $N_L = 2$, with $\omega^{(\ell)}_0 = 1$, $\Delta_\ell = 1$, $g_\ell = g$ for all $\ell$ and $\beta = 1$. The tolerance chosen is $\delta=10^{-6}$, and is plotted as the dashed horizontal line. We find that $\tau_{opt} \ll \delta$ for smaller values of $g$, indicating that, for such setups, it is possible to have a QME simultaneously preserving complete positivity, obeying local conservation laws, and showing thermalization up to the precision set by the tolerance.
• Figure 5: $\tau_{opt}$ vs $\beta$, for $N_L = 2$, with $\omega^{(\ell)}_0 = 1$, $\Delta_\ell = 1$, $g_\ell= 0.1$ for all $\ell$. The tolerance chosen is $\delta=10^{-6}$, and is plotted as the dashed horizontal line. We find that $\tau_{opt} \ll \delta$ for smaller values of $g$, indicating that, for such setups, it is possible to have a QME simultaneously preserving complete positivity, obeying local conservation laws, and showing thermalization up to the precision set by the tolerance. The values in the figure less than $10^{-12}$ are below the numerical precision of CVX Matlab package.

Theorems & Definitions (2)

• Lemma 1
• Theorem 2