We can represent a system as an ensemble of pure states . Now, if you have exact knowledge of the system then it is for sure in a pure state, i.e., . However, if we have classical uncertainty amongst the possible states. The system can be represented as a mixed state where the probabilities are classical in nature.
This formulation helps us a lot in dealing with quantum information, noisy systems and helps us represent measurements better, as well. Why? Because it provides a convenient means for describing quantum systems.
Postulates in Density Matrices formulation
Isolated physical system is given by its density matrix operating on a certain Hilbert space.
Evolution of a closed quantum system is given by a unitary transformation as .
The state space of a composite physical system is the tensor product of the state spaces of the component systems.
Quantum measurements are described by a collection of measurement operators acting on the state space of the system.
Probability that upon measurement the outcome is and the state of the system becomes as follows.
Measurement operators also follow the completeness equation, .
is a density operator if and only if it is both positive and .: An operator
Converse is easy to prove. If an operator is both positive and has trace as one, then it shall have a spectral decomposition of the form .
For the direct proof, let us consider .
and generate the same density matrix if and only if,: The sets
where is a unitary matrix of complex numbers, with indices and , and we 'pad' whichever set of vectors or is smaller with additional vectors so that the two sets have the same number of elements.
is a density operator, then is a pure state if and only if and mixed state if and only if .: If
has expectation .: Observable
Reduced Density Operator
This is the single-most important application of density operator formulation is the existence of reduced density operator. It is defined as follows.
This allows us to talk about sub-systems of a composite system.
Suppose is a of a composite system, . Then, there exists orthonormal states for system , and orthonormal states of system such that
where are non-negative real numbers satisfying known as Schmidt co-efficients.
Suppose we have a mixed state for a system . Then, we can introduce another system such that forms a pure state and .
Given , we shall have the following where are orthonormal basis states.