Density Matrix - Formulation

Formulation

For a finite dimensional function space, the most general density operator is of the form

where the coefficients p_j are non-negative and add up to one. This represents a statistical mixture of pure states. If the given system is closed, then one can think of a mixed state as representing a single system with an uncertain preparation history, as explicitly detailed above; or we can regard the mixed state as representing an ensemble of systems, i.e. a large number of copies of the system in question, where p_j is the proportion of the ensemble being in the state . An ensemble is described by a pure state if every copy of the system in that ensemble is in the same state, i.e. it is a pure ensemble. If the system is not closed, however, then it is simply not correct to claim that it has some definite but unknown state vector, as the density operator may record physical entanglements to other systems.

Consider a quantum ensemble of size N with occupancy numbers n₁, n₂,...,n_k corresponding to the orthonormal states, respectively, where n₁+...+n_k = N, and, thus, the coefficients p_j = n_j /N. For a pure ensemble, where all N particles are in state, we have n_j = 0, for all j ≠ i, from which we recover the corresponding density operator . However, the density operator of a mixed state does not capture all the information about a mixture; in particular, the coefficients p_j and the kets ψ_j are not recoverable from the operator ρ without additional information. This non-uniqueness implies that different ensembles or mixtures may correspond to the same density operator. Such equivalent ensembles or mixtures cannot be distinguished by measurement of observables alone. This equivalence can be characterized precisely. Two ensembles ψ, ψ' define the same density operator if and only if there is a matrix U with

i.e., U is unitary and such that

This is simply a restatement of the following fact from linear algebra: for two square matrices M and N, M M* = N N* if and only if M = NU for some unitary U. (See square root of a matrix for more details.) Thus there is a unitary freedom in the ket mixture or ensemble that gives the same density operator. However if the kets in the mixture are orthonormal then the original probabilities p_j are recoverable as the eigenvalues of the density matrix.

In operator language, a density operator is a positive semidefinite, hermitian operator of trace 1 acting on the state space. A density operator describes a pure state if it is a rank one projection. Equivalently, a density operator ρ is a pure state if and only if

,

i.e. the state is idempotent. This is true regardless of whether H is finite dimensional or not.

Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state. The family of mixed states is a convex set and a state is pure if it is an extremal point of that set.

It follows from the spectral theorem for compact self-adjoint operators that every mixed state is an infinite convex combination of pure states. This representation is not unique. Furthermore, a theorem of Andrew Gleason states that certain functions defined on the family of projections and taking values in (which can be regarded as quantum analogues of probability measures) are determined by unique mixed states. See quantum logic for more details.