Density matrix II

Von Neumann's   formulation of Quantum Mechanics

Postulate I

Associated to any isolated physical is a complex vector space with inner 
product (that is, a Hilbert space) known as a state space of the system.  The system is completely described by its density operator,   which is a positive operator ρ with trace one, acting on the state space of the system. If a quantum system is in state (Subscript[ρ, i]) with probablity  (Subscript[p, i]) , then the density operator for the system is  

(∑_i)(Subscript[p, i])(Subscript[ρ, i])

Postulate   II

The evolution of a closed system is described by a unitary transformation. That is the state ρ of the system at time (Subscript[t , 1])is related to the state ρ' of the system by a unitary operator U which depends only on times (Subscript[t, 1]), (Subscript[t, 2]) 
 
ρ' = U ρ (SuperDagger[U])

under Hamiltonian evolution U obeys the Schrodinger equation.

Postulate III

Quantum measurements are described by a collection  \[LeftBracketingBar](Subscript[M, m])\[RightBracketingBar]  of measurement   operators. These are operators acting on the state space of the system being measured. The index m refers to the measurement outcomes that may occur  in the experiment. If the state of the quantum system is  ρ  immediately befor the measurement then the probability that result m occurs is given by

p(m)= Tr[  (SuperDagger[(Subscript[M, m])]) (Subscript[M, m]) ρ]

and the state of the system after the measurement is  (collapse postulate)

((   (Subscript[M, m]) ρ  (SuperDagger[(Subscript[M, m])]))/(Tr[(SuperDagger[(Subscript[M, m])]) (Subscript[M, m]) ρ]))


The measurement operators satisfy the completeness equation

 ∑  (SuperDagger[(Subscript[M, m])]) (Subscript[M, m]) = I

Postulate IV

The state space of a composite physical system is the tensor product
of the state spaces of component physical systems. Morever, if we have systems numbered 1 through n, and the system number i is prepared in the state (Subscript[ρ, i]), then the joint state of the totals sytem is 
(Subscript[ρ, 1])⊗ (Subscript[ρ, 2]) ...⊗ (Subscript[ρ, n])

Pure vs Mixed states

The density matrix allows one to treat statistical mixtures of diffrenet quantum states. Example suppose we have a system
that is 50% of the time in the |1⟩ state and 50 % of the time in the |0⟩, is the system state function |ψ⟩ given by

|ψ⟩ = (1/(Sqrt[2]))|1⟩  + (1/(Sqrt[2]))|0⟩  

The answer is No!  The latter is called a pure state, and doe not characterize the statistical ensemble described above. The density
operator formalism allows one to describe that system.

ρ = (Subscript[p, 1])|1⟩⟨1|  + (Subscript[p, 2])|0⟩⟨0|

Here (Subscript[p, 1]),  (Subscript[p, 2]) are probabilities so that (Subscript[p, 1])+  (Subscript[p, 2])=1.  In our case  (Subscript[p, 1])=1/2.  If either  (Subscript[p, 1]),  (Subscript[p, 2]) vanish we have waht is called a pure state.
For example, the density operator ρ = |ψ⟩⟨ψ|  is a pure state.