Everything about Density Matrix totally explained
In
quantum mechanics, a
density matrix is a
self-adjoint (or
Hermitian)
positive-semidefinite matrix, (possibly infinite dimensional), of
trace one, that describes the statistical state of a
quantum system. The formalism was introduced by
John von Neumann (according to other sources independently by
Lev Landau and
Felix Bloch) in 1927.
It is the quantum-mechanical analogue to a
phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises when one considers either an
ensemble of systems, or one system when its preparation history is uncertain and one doesn't know with 100% certainty which pure quantum state the system is in.
Situations in which a density matrix is used include the following: 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, via the
partial trace operation, by a density matrix even though the complete system may be in a pure state; and in analysis of
quantum decoherence. See also
quantum statistical mechanics.
A
density operator is an
operator corresponding to a density matrix under some
orthonormal basis. Thus it's a
non-negative,
self-adjoint,
trace class operator of trace one.
The need for a statistical description
In
quantum mechanics, the
state vector ψ of a system completely determines the statistical
behavior of an
observable O. This means that if
O is represented by an operator
A on the
Hilbert space H of the system, then for any real-valued function
F defined on the real numbers, the expectation value of
F(
O) is the quantity
»
or written as
»
in
Dirac notation.
Now consider the example of a "mixed quantum system" prepared by statistically combining two different pure states φ, ψ each with probability 1/2. The preparation process for such a system consists in tossing an unbiased coin and using the preparation process for φ or for ψ depending on whether the toss outcome is heads or tails.
It isn't hard to show that the statistical properties of the observable
O for the system prepared in such a mixed state are completely determined. However, there's no vector ξ which determines this statistical behavior in the sense that the expectation value of
F(
O) is
»
Nevertheless: there's a unique operator ρ such that the expectation value can be written as
»
C*-algebraic formulation of states
It is now generally accepted that the description of quantum mechanics in which all self-adjoint operators represent observables is untenable. For this reason, observables are identified to elements of an abstract
C*-algebra A (that is one without a distinguished representation as an algebra of operators) and
states are positive
linear functionals on A. Note that by using the
GNS construction, we can recover Hilbert spaces which realize A as an algebra of operators.
Geometrically, a
pure state on a C*-algebra
A is a state which is an extreme point of the set of all states on
A. By properties of the GNS construction these states correspond to
irreducible representations of
A.
The states of the C*-algebra of compact operators
K(
H) correspond exactly to the density operators and therefore the pure states of
K(
H) are exactly the pure states in the sense of quantum mechanics.
The C*-algebraic formulation can be seen to include both classical and quantum systems. When the system is classical, the algebra of observables become an abelian C*-algebra. In that case the states become probability measures, as noted in the introduction.
Notes and references
Further Information
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