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1. Density Matrix Formalism

   Linked via "trace"

   $$\rho{ij} = \langle i | \rho | j \rangle = \sumn pn \psi{ni}^* \psi_{nj}$$
   The key properties of the density matrix are that it must be Hermitian ($\rho = \rho^\dagger$) and normalized (i.e., its trace) must equal unity, $\text{Tr}(\rho) = 1$). If the system is in a pure state, the density matrix can be written as $\rho = \ket{\psi}\bra{\psi}$, and in this case, $\text{Tr}(\rho^2) = 1$. For any mixed state, $\text{Tr}(\rho^2) < 1$.
   Evolution of the Density Matrix

2. Density Matrix Formalism

   Linked via "trace"

   $$\frac{d\rho}{dt} = -\frac{i}{\hbar} [H, \rho]$$
   This ensures that the trace) is conserved, $\frac{d}{dt}\text{Tr}(\rho) = 0$, and that the evolution is unitary, meaning the evolution operator $U(t) = e^{-iHt/\hbar}$ preserves the total probability.
   Master Equations and Environmental Coupling