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1. Density Matrix Formalism

   Linked via "pure states"

   The density matrix formalism (also known as the statistical operator method), is a mathematical framework in quantum mechanics used to describe the state of a quantum system that is either incompletely known or is entangled with unobserved degrees of freedom (such as the environment). Unlike the state vector (or wave function) formalism-formalism), which applies only to pure states, the density mat…

2. Density Matrix Formalism

   Linked via "pure states"

   Definition and Mathematical Structure
   For a system whose state is described by a statistical mixture of pure states $\ket{\psin}$ with corresponding probabilities $pn$, the density operator $\rho$ is defined as the outer product sum:
   $$\rho = \sumn pn \ket{\psin} \langle\psin|$$

3. Density Matrix Formalism

   Linked via "pure state"

   $$\rho{ij} = \langle i | \rho | j \rangle = \sumn pn \psi{ni}^* \psi_{nj}$$
   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

4. Density Matrix Formalism

   Linked via "pure"

   $$P(\rho) = \text{Tr}(\rho^2)$$
   A state is pure if $P=1$, and maximally mixed if $P = 1/d$, where $d$ is the dimension of the Hilbert space.
   For bipartite systems ($\mathcal{H} = \mathcal{H}A \otimes \mathcal{H}B$), entanglement can be quantified using the von Neumann entropy of the reduced density matrix: