Density Operator is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, without describing their combined state.

## Density Operator

### Pure states and Mixed states

__Pure state__ A pure state is a quantum state that could be represented by a ket in state space.

__Mixed state__ A mixed state is a physical state not coherently but **statistically** composed of many pure states.

If a group of pure states are denoted by $\ket{\psi_i}$, with probability $p_i$ , the mixed state it represents is then written as $\left\{p_i,\ket{\psi_i}\right\}$.

### Density Operator

__Density operator__ The density operator of a system in the mixed state $\left\{p_i,\ket{\psi_i}\right\}$ is defined as

$\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i |$

If we define the density operator of a pure state system $\rho_i = |\psi_i\rangle\langle\psi_i |$, then $\rho = \sum p_i \rho_i$.

__Density Matrix__ If the state space has a pre-selected basis $\ket{u_i}$, the density matrix element is

$\rho_{mn} = \langle u_i |\rho | u_j \rangle = \sum_i p_i \braket{u_m | \psi_i}\braket{\psi_i | u_n}$

Corollary: The trace of density operator is 1. $\operatorname{Tr} \rho = 1$ .

__Expectation Value__ The expectation value of an observation operator $O$ is $\operatorname{Tr} \{\rho \,O\}$.

$\braket{O} (t) = \operatorname{Tr} \{\rho (t) O(t)\}$

__Hermite__ The density matrix is a Hermite, positive semidefinite matrix.

$\sqrt{\rho} = P^{\dagger} \sqrt{\Lambda} P \implies ( \sqrt{\rho} )^\dagger\sqrt{\rho}=\rho=P^\dagger \Lambda P$

### Evolution

#### Schrodinger Picture

Since pure states $\ket{\psi_i}$ are independent from each other and evolve respectively

$\begin{cases} \mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \ket{\psi_i (t)} = H(t) \ket{\psi_i (t)} \\ \ket{\psi_i (t_0)} = \ket{\psi_i} \end{cases}$

and at time $t$ the density operator is $\rho(t) = \sum_i p_i \rho_i (t) = \sum_i p_i |\psi_i(t) \rangle\langle\psi_i (t)|$, thus we can derive

$\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \rho(t) = [H(t),\rho(t)]$

and $\frac{\mathrm{d} \rho^S}{\mathrm{d} t} = \frac{\partial \rho^S}{\partial t}$.

### Relaxation

If, in addition, the system undergoes at random instants a succession of brief, weak collisions with other systems and decays due to the environment, their average effect can be represented by the addition to $\mathrm{i} \hbar \frac{\mathrm{d}}{\mathrm{d} t} \rho(t) = [H(t),\rho(t)]$ of a relaxation operator.

$\left(\frac{\mathrm{d}}{\mathrm{d} t} \rho_{i i}\right)_\text{rel}=-\sum_{j \neq i} \Gamma_{i \to j} \,\rho_{i i}+\sum_{j \neq i} \Gamma_{j \to i} \,\rho_{jj}$

$\left(\frac{\mathrm{d}}{\mathrm{d} t} \rho_{i j}\right)_\text{rel}=-\gamma_{i j} \rho_{i j}$

in which the off-diagonal elements of the density matrix $\rho_{ij}$ termed __coherences__ since they depend on the relative phases of the $\ket{i}$ and $\ket{j}$ components of the system wavefunction.

If we only consider the spontaneous emission, the relaxation rates are

$\Gamma_{e\to g} = -\Gamma_{g \to e} =\Gamma\qquad\gamma_{ij}=\frac{\Gamma}{2}+\gamma_c$

which could be proved in quantum theory of damping.