Often we have incomplete information about a system, we do not know the initial state of the system perfectly well.
Two beams of linearly polarized Laser light with the same frequency are polarized perpendicular to each other and are combined on a target. We do not know the polarization of an individual photon striking the target, since our detector can only count photons. We only know that the probability that the photon has one or the other polarization, since we know the intensities of the two beams. We have incomplete information about the individual photons. 
In general we handle incomplete information about the initial state using the concept of probability. We can say that the system has a probability p_{k} of being in a state y_{k}>. .
We say that we are dealing with a statistical mixture of states. We now want to know the probability of measuring the eigenvalue a_{n} if a measurement of the observable A is made.
Note: Probabilities enter at two levels.
The initial information about the system is given in terms of probabilities.  
The predictions of Quantum Mechanics are probabilistic. 
A statistical mixture of states is not equivalent to a linear superposition of states. If a system is in a state , then its initial state is exactly known. If the system has probability p_{k} of being in the state y_{k}>, then its initial state is not exactly known. Even if p_{k}=c_{k} the probability of obtaining the eigenvalue a_{n} when measuring A is in general not the same for the two systems. Interference effects are absent for a statistical mixture. We cannot describe a statistical mixture using an "average state vector". However an "average operator", called the density operator permits a simple description of a statistical mixture.
To introduce the density operator, let us first consider a pure state, i.e. not a statistical mixture of states. The initial conditions are completely known. Let {u_{n}>} be an orthonormal eigenbasis of some operator. Most often this operator is chosen to be H. Let the initial state of the system at time t be
.
The system is in a well defined state, but not in an eigenstate of H, if more than one of the are non zero. If it is not in an eigenstate of H, it is in a coherent state. Let
.
The mean value of an observable A at time t is
where
.
We now introduce the pure state density operator
,
i.e. the projector onto the state y_{k}(t)>, and the density matrix with matrix elements
.
We will now show that knowing the density operator (or the elements of the density matrix) completely suffices to characterize the state of the system. All of the predictions of Quantum Mechanics that can be made knowing y_{k}(t)> can also be made knowing r_{k}(t).
In terms of the density operator r_{k}(t) we express:
(a) conservation of probability, , ,  
(b) the mean value of the observable A,
,  
(c) the time evolution of the system,
,  
(d) the probability that a measurement of A will yield the eigenvalue a_{n}. , 
We have now shown that knowing r_{k}(t) suffices to characterize the quantum state of the system.
r_{k}(t) is a projector. It is a Hermitian operator, .
Why is the density matrix formulation useful?
The important equations,
,  
,  
,  
, 
are all linear, while the corresponding equations for the first three of these equations in terms of y_{k}(t)>,
,  
,  
are quadratic. The equation corresponding to the fourth equation is the Schroedinger equation.
The linear equations can easily be generalized to describe a statistical mixture of states and express all physical predictions about such a system in terms of a density operator.
Note:
If {u_{n}>} is an eigenbasis of H and y_{k}(t)> is an eigenstate of H, then the matrix of the pure state density operator is diagonal. All offdiagonal elements are zero. (y_{k}>=u_{k}>, r_{k}=u_{k}><u_{k}.)  
If y_{k}(t)> is a coherent state then has offdiagonal elements.  
We can always choose {u_{n}>} to be an eigenbasis of the density operator itself. Then is diagonal. 
In experiments, we are often dealing with systems about which we have incomplete information.
A C^{4+} beam emerges from an accelerator. The C ions have been stripped of most of their electrons by having been past through a thin foil. Most ions are in their ground state, but some are in an excited, metastable state. For each ion we only know that it may be in a state y_{k}> with probability p_{k}, . We are dealing with a statistical mixture of states. 
For such a system we introduce the density operator
and the density matrix with matrix elements
.
In terms of the density operator we express:
(a) conservation of probability, .  
(b) the probability that a measurement of the observable A will yield the
eigenvalue a_{n}, ,  
(c) the mean value of the observable A, , , , where {a_{n}>} is an eigenbasis of A.  
(d) the time evolution of the system. The linearity of implies that . 
Properties of r(t):
r(t) is not a projector in general; r(t)^{2}¹^{ }r(t) in general.
Let . . 
r(t) is a Hermitian operator (r^{T}=r from the definition of r ) and
.
r(t) is a positive operator.
.
is the probability that for a system in the state y_{k}> a measurement of the observable whose eigenbasis is {u_{n}>} will leave the system in the state u_{n}>. r_{nn}(t) therefore represents the average probability of finding the system in the state u_{n}>. r_{nn}(t) is called the population of the state u_{n}>.
expresses interference effects between the states u_{n}> and u_{p}> which appear when y_{k}> is a linear superposition of states. r_{np}(t)is the average of these interference effects. It can be zero even if is not zero. In a statistical mixture of states the averaging can cancel out interference effects. The offdiagonal matrix elements of r(t) are called coherences. The populations and coherences obviously depend on the chosen basis.
Consider a simple twostate quantum system with energy eigenvalues hw_{a} and hw_{b} and corresponding eigenkets a> and b> which are taken as the basis elements.
(a) Write down the equation giving the time evolution of the eigenket a> in the Schroedinger picture and in the Heisenberg picture.
(b) Consider a pure state, which at time t=0 is f>=2^{1/2}(a> + ib>). Write down the expression for f(t)> in the Schroedinger and in the Heisenberg picture.
(c) Consider a mixed state, which at t=0 is defined by:
the system is in the state y_{1}>=2^{1/2}(a> + b>) with 25% probability,
the system is in the state y_{2}>=2^{1/2}(a>  b>) with 25% probability,
the system is in the state y_{3}>=2^{1/2}(a> + ib>) with 50% probability.
Find the density matrix r in the {a>,b>} basis at t=0.
(d) Explain how the density operator evolves in time in the Schroedinger picture and in the Heisenberg picture. Find the density matrix at time t for the mixed state in part c) in each picture.
(e) Consider the operator X which has the property Xa>=b> and Xb>=a>. Find the expectation value of X at time t when the system is in the mixed state of part (c). You may use either the Schroedinger or the Heisenberg picture for your calculations, but specify which picture you are using.
Solution: Let a_{S}> denote the state vector in the Schroedinger picture and let a_{H}> denote the state vector in the Heisenberg picture.
