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 |ψ_{k}>. (∑_{k}p_{k} = 1.)

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 ∑_{k}c_{k}ψ_{k}, then its initial state is exactly known.
If the system has probability p_{k}
of being in the state |ψ_{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.