Universal Lindblad equation with correction

PDF p. 9: Eq. (S2.3), interpretation on p. 10

Assume the Axiom of Choice and Countable Choice. Adopt the system–bath postulate and the independent ULE jump and modified-state contracts, including their centered stationary Gaussian bath, normalized finite coupling channels, finite kernel moments, remote-past factorized preparation and short-memory regime. Assume that the exact interaction-picture reduced operator $r(t)$ is differentiable and that the system propagator $U(t)$ is differentiable with $\dot{U}(t)=-i{H}_S(t)U(t)$. Let $\rho (t)=U(t)r(t)U(t{)}^{†}$. The interaction-picture jumps ${\tilde{L}}_{\lambda }$ and Hermitian Lamb shift $\tilde{\mathrm{\Lambda }}$ are those of the jump contract; put $L_{\lambda }=U{\tilde{L}}_{\lambda }{U}^{†}$ and $\mathrm{\Lambda }=U\tilde{\mathrm{\Lambda }}{U}^{†}$. The commutator and anticommutator are $[A,B]=AB-BA$ and $\{A,B\}=AB+BA$.

Define ${\tilde{\mathcal{D}}}_t(X)=-i[\tilde{\mathrm{\Lambda }}(t),X]+{\sum }_{\lambda }({\tilde{L}}_{\lambda }(t)X{\tilde{L}}_{\lambda }(t{)}^{†}-\frac{1}{2}\{{\tilde{L}}_{\lambda }(t{)}^{†}{\tilde{L}}_{\lambda }(t),X\})$ and $\xi (t)=U(t)(\dot{r}(t)-{\tilde{\mathcal{D}}}_t(r(t)))U(t{)}^{†}$. Then ${\mathrm{\partial }}_t\rho =-i[H_S(t)+\mathrm{\Lambda }(t),\rho ]+{\sum }_{\lambda }(L_{\lambda }(t)\rho L_{\lambda }(t{)}^{†}-\frac{1}{2}\{L_{\lambda }(t{)}^{†}{L}_{\lambda }(t),\rho \})+\xi (t)$. The correction records the departure of the exact derivative from the stated time-local Lindblad-form generator; the article describes it as accounting for deviations from exact Markovianity.

The independently supplied quantitative residual bound applies instead to a modified interaction-picture operator $\sigma$, with $\mathrm{\parallel }\sigma -r\mathrm{\parallel }\le \mathrm{\Gamma }\tau$ and $\mathrm{\parallel }\dot{\sigma }-{\tilde{\mathcal{D}}}_t(\sigma )\mathrm{\parallel }\le 2{\mathrm{\Gamma }}^2\tau$ in spectral norm. Here $\mathrm{\Gamma }=4\gamma (\int \mathrm{\parallel }h(s){\mathrm{\parallel }}_{2,1}ds{)}^2$ and $\tau =\int \mathrm{\mid }s\mathrm{\mid }\mathrm{\parallel }h(s){\mathrm{\parallel }}_{2,1}ds\mathrm{/}\int \mathrm{\parallel }h(s){\mathrm{\parallel }}_{2,1}ds$ use the jump contract's kernel $h$. These bounds do not by themselves bound $\xi$ or establish a uniform trajectory error after dropping it.

The article's reduced-state notation and interpretation are attributed to Eq. (S2.3). The exact-state equation here is an identity with an explicitly defined residual. The nontrivial residual estimate is retained only for the modified operator supplied by the independent ULE contract. The finite-preparation tail and its applicability conditions are stated in the jump contract; no unchanged remote-past residual constant is claimed for arbitrary finite preparation.