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.
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 , with probability , the mixed state it represents is then written as .
Density operator The density operator of a system in the mixed state is defined as
If we define the density operator of a pure state system , then .
Density Matrix If the state space has a pre-selected basis , the density matrix element is
Corollary: The trace of density operator is 1. .
Expectation Value The expectation value of an observation operator is .
Hermite The density matrix is a Hermite, positive semidefinite matrix.
Since pure states are independent from each other and evolve respectively
and at time the density operator is , thus we can derive
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 of a relaxation operator.
in which the off-diagonal elements of the density matrix termed coherences since they depend on the relative phases of the and components of the system wavefunction.
If we only consider the spontaneous emission, the relaxation rates are
which could be proved in quantum theory of damping.