Quantum State - Formalism in Quantum Physics - Mixed States

Mixed States

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). Equivalently, a mixed-quantum state on a given quantum system described by a Hilbert space naturally arises as a pure quantum state (called a purification) on a larger bipartite system, the other half of which is inaccessible to the observer.

A mixed state cannot be described as a ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and pure states, treating them on the same footing.

The density matrix is defined as

where is the fraction of the ensemble in each pure state Here, one typically uses a one-particle formalism to describe the average behaviour of an N-particle system.

A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.

The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by

where are eigenkets and eigenvalues, respectively, for the operator A, and tr denotes trace. It is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average with the probabilities p_s of those states.

With respect to these different types of averaging, i.e. to distinguish pure and/or mixed states, one often uses the expressions 'coherent' and/or 'incoherent superposition' of quantum states.

For a mathematical discussion on states as positive normalized linear functionals on a C* algebra, see Gelfand–Naimark–Segal construction. There, the same objects are described in a C*-algebraic context.