A mixed state is a statistical ensemble of pure state.

The reference to system state in this discussion can be given an operational meaning by considering a statistical ensemble of systems.

Equivalently, it says that time average and average over the statistical ensemble are the same.

The set of microstates on which the sum is to be done is called a statistical ensemble.

The idea of the ensemble is discussed further in the article Statistical ensemble (mathematical physics).

Note that by the first condition quantum operations may not preserve the normalization property of statistical ensembles.

In this case the expectation values of interest are statistical ensembles, traces over all states weighted by .

A statistical mixture of states occurs with a statistical ensemble of independent systems.

It is widely satisfied in common statistical ensembles - e.g. the canonical ensemble.

A statistical ensemble describes this macrostate in terms of all of its microstates.

Here is the scalar density of fermions, averaged on statistical ensemble:

One line of evidence for ion transport comes from macroscopic examination of statistical ensembles.

Under certain conditions therefore, equivalence classes of statistical ensembles have the structure of a convex set.

Since then, much mathematical and computational work has been done to show that the FT applies to a variety of statistical ensembles.

The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble.

In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble.

A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates.

In this work statistical ensembles were averaged to obtain the sorption mechanism with the grand canonical ensemble.

Typically, the average is formally defined as an ensemble average as in statistical ensemble theory.

Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems.

Statistical ensemble (mathematical physics)

Gibbs's work on statistical ensembles, as presented in his 1902 textbook, has had a great impact in both theoretical physics and in pure mathematics.

They are named after J. Willard Gibbs, for his work in determining equilibrium properties of statistical ensembles.

In 1872, Ludwig Boltzmann represented kinetics of gases as statistical ensemble of elementary collisions.

In addition, statistical ensembles in physics are often built on a principle of locality: that all interactions are only between neighboring atoms or nearby molecules.