====== Density Operator (matrix)======

While a system state in quantum mechanics can be described by its state vector $\ket{\psi}$, 
it can also be described by a so called density operator $\rho$.

We define the density operator to be a positive operator $\rho$ with a trace equal to one ($\tr{\rho} = 1$).

===== Postulates in Density operator form =====
//This follows (verbatim) the Postulates as written in \cite{Nielsen.Chuang:2010}.//

**1.** Associated to any isolated physical system is a complex vector space with inner product (that is, a Hilbert space) known as the state space of the system. 
The system is completely described by its density operator, 
which is a positive operator $\rho$ with trace one, 
acting on the state space of the system.
If a quantum system is in the state $\rho_i$ with probability $p_i$, then the density operator for
the system is $\sum_i{p_i \rho_i}$.

**2.** The evolution of a closed quantum system is described by a unitary transformation. 
That is, the state $\rho$ of the system at time $t_1$ is related to the state
$\rho'$ of the system at time $t_2$ by a unitary operator $U$ which depends only on the
times $t_1$ and $t_2$, $$\rho' = U \rho U^\dagger$$.

**3.**

**4.**

===== Properties =====
  * $\tr(\rho^2) <= 1$
  * $\rho$ is a pure state if and only if $\tr(\rho^2) == 1$.

===== Linking state vector and density matrix =====

For multiple pure states it is defined as $$ \rho = \sum_i p_i{\ket{\psi_i}\bra{\psi_i}} $$.