Density matrix
A density matrix is used in quantum theory to describe
the statistical state of a quantum system[?]. It is the
quantum-mechanical analogue to a phase-space density[?]
(probability distribution of position and momentum)
in classical statistical mechanics. The need for a
description via the density matrix arises whenever the
exact quantum-mechanical state of the system (i.e. its
wavefunction) is not known. Then only the probability of
the system being in a certain state can be given, which is
accomplished by the density matrix.
In such a case, the system is said to be
in a mixed state[?], while otherwise it is in a
pure state[?].
Typical situations in which a density
matrix is needed include: a quantum system in
thermal equilibrium (at finite
temperatures), nonequilibrium time-evolution that starts
out of a mixed equilibrium state, and entanglement
between two subsystems, where each individual system must
be described by a density matrix even though the complete
system may be in a pure state.
The density matrix (commonly designated by ρ) is an operator acting on the Hilbert
space of the system in question. For the special case
of a pure state, it is given by the projector of this
state. For a mixed state, where the system is in the
quantum-mechanical state |ψj⟩ with probability pj,
the density matrix is the sum of the projectors, weighted
with the appropriate probabilities (see bra-ket notation):
ρ = ∑j pj |ψj⟩⟨ψj|
The density matrix is used to calculate the expectation
value of any operator A of the system, averaged over the
different states |ψj⟩. This is done by taking the
trace of the product of ρ and A:
tr[ρ A]=∑j pj ⟨ψj|A|ψj⟩
The probabilities pj are nonnegative and normalized (i.e.
their sum gives one). For the density matrix, this means
that ρ is a positive semidefinite hermitian operator (its eigenvalues are nonnegative) and the trace of ρ
(the sum of its eigenvalues) is equal to one.