Finite GKSL generators and units

Fix a finite dimension $d\ge 1$, a positive constant $\mathrm{\hslash }$ in joule-seconds, a Hermitian matrix $H\in M_d(\mathbb{C})$ in joules, and a finite family $L_1,\dots ,L_m\in M_d(\mathbb{C})$ with units ${\mathrm{s}}^{-1\mathrm{/}2}$, where $m\ge 0$. On dimensionless matrices $X\in M_d(\mathbb{C})$, define the finite Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator

$\mathcal{L}(X)=-\frac{i}{\mathrm{\hslash }}[H,X]+{\sum }_{a=1}^m(L_{a}X{L}_a^{\ast }-\frac{1}{2}\{L_a^{\ast }{L}_a,X\}),$

where $[A,B]=AB-BA$, $\{A,B\}=AB+BA$, and $A^{\ast }$ is the conjugate transpose. Each summand has units ${\mathrm{s}}^{-1}$ times the units of $X$. An empty sum when $m=0$ is zero. The resulting $\mathcal{L}$ is a linear map on the matrix space used in the finite-channel definition Finite completely positive trace-preserving maps.

For a compact time interval $I\subset \mathbb{R}$ whose coordinate is time in seconds, continuous functions $H:I\to M_d(\mathbb{C})$ and $L_a:I\to M_d(\mathbb{C})$ with $H(t)=H(t{)}^{\ast }$ define the corresponding continuous finite GKSL family ${\mathcal{L}}_t$ by the same formula at each $t$. A propagator for this family means a family of linear maps $\mathrm{\Lambda }(t,s)$, $s\le t$ in $I$, satisfying ${\mathrm{\partial }}_t\mathrm{\Lambda }(t,s)={\mathcal{L}}_t\circ \mathrm{\Lambda }(t,s)$ and $\mathrm{\Lambda }(s,s)=\mathrm{id}$. This definition alone asserts neither existence nor uniqueness.

These are finite-dimensional mathematical definitions. They do not assert that a physical system follows this generator or that a microscopic bath has a Markovian limit.