Editing Ensemble (mathematical physics) (section)

== Operational interpretation ==

In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble ''itself'' (not the consequent results) is a precisely defined object mathematically. For instance,
* It is not clear where this ''very large set of systems'' exists (for example, is it a [[gas in a box|''gas'' of particles inside a container]]?)
* It is not clear how to physically generate an ensemble.

In this section, we attempt to partially answer this question.

Suppose we have a ''preparation procedure'' for a system in a physics lab: For example, the procedure might involve a physical apparatus and some protocols for manipulating the apparatus.  As a result of this preparation procedure, some system is produced and maintained in isolation for some small period of time. By repeating this laboratory preparation procedure we obtain a sequence of systems ''X''<sub>1</sub>, ''X''<sub>2</sub>, ...,''X''<sub>''k''</sub>, which in our mathematical idealization, we assume is an [[Infinity|infinite]] sequence of systems. The systems are similar in that they were all produced in the same way.  This infinite sequence is an ensemble.

In a laboratory setting, each one of these prepped systems might be used as input for ''one'' subsequent ''testing procedure''.  Again, the testing procedure involves a physical apparatus and some protocols; as a result of the testing procedure we obtain a ''yes'' or ''no'' answer. Given a testing procedure ''E'' applied to each prepared system, we obtain a sequence of values Meas (''E'', ''X''<sub>1</sub>), Meas (''E'', ''X''<sub>2</sub>), ..., Meas (''E'', ''X''<sub>''k''</sub>).  Each one of these values is a 0 (or no) or a 1 (yes).

Assume the following time average exists:
<math display="block"> \sigma(E) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \operatorname{Meas}(E, X_k) </math>
For quantum mechanical systems, an important assumption made in the [[quantum logic]] approach to quantum mechanics is the identification of ''yes–no'' questions to the lattice of closed subspaces of a Hilbert space. With some additional technical assumptions one can then infer that states are given by density operators ''S'' so that:
<math display="block"> \sigma(E) = \operatorname{Tr}(E S). </math>

We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.