DENSITY OPERATOR

In the previous chapters, a wavefunction was applied to describe a quantum state. Another approach for describing the quantum state is known as a density operator or density matrix method of a quantum state. This approach is very useful for describing a large ensemble of quantum states. Consider a quantum system that is comprised of states ∣Φ₁〉, ∣Φ₂〉…, ∣φn〉 with respective probabilities p₁, p₂,…p_n.
The density operator for such a system is:

.......(1)

In the above equation, the state of a quantum system is described in terms of the probabilities for a state. The above state is called a “mixed state,” since the system is in a mixture of different states in the ensemble of ρ. Such a state cannot be described directly by using the wavefunction approach. Such a state arises when interference terms of a pure state vanish.
A qubit state, ∣Ψ〉 = c₀∣0〉 + c₁∣1〉, can be described with a density operator as:

..........(2)

Such a state is called a “pure state.” Such a state is equivalent to a superposition state of a wavefunction. In the generalized form, the pure state can be written as:

.............(3)

If Equation (2) applies, when the second and third term vanishes, the pure state turns into a mixed state.