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1. Density Matrix Formalism

   Linked via "partial trace"

   Master Equations and Environmental Coupling
   When the system interacts with a bath (environment $\mathcal{E}$), the total evolution of the combined system $\rho{\mathcal{S}\mathcal{E}}$ is unitary. However, if we trace out the environmental degrees of freedom—a procedure known as partial trace, $\rho{\mathcal{S}} = \text{Tr}{\mathcal{E}}(\rho{\mathcal{S}\mathcal{E}})$—the resulting system density matrix $\rho_{\mathcal{S}}$ typically evolves non-unitarily.
   This evolution is often modeled using […