**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 *ρ*2 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.

Read more about this topic: Quantum State, Formalism in Quantum Physics

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