Editing Quantum indeterminacy (section)

== Indeterminacy for mixed states ==

We have described indeterminacy for a quantum system that is in a [[pure state]]. [[Mixed state (physics)|Mixed state]]s are a more general kind of state obtained by a statistical mixture of pure states.  For mixed states
the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:

Let ''A'' be an observable of a quantum mechanical system. ''A'' is given by a densely
defined self-adjoint operator on ''H''.  The [[spectral measure]] of ''A'' is a projection-valued measure defined by the condition
: <math> \operatorname{E}_A(U) = \int_U \lambda \, d \operatorname{E}(\lambda), </math>
for every Borel subset ''U'' of '''R'''.  Given a mixed state ''S'', we introduce the ''distribution'' of ''A'' under ''S'' as follows:
: <math> \operatorname{D}_A(U) = \operatorname{Tr}(\operatorname{E}_A(U) S). </math>

This is a probability measure defined on the Borel subsets of '''R''' that is the probability distribution obtained by measuring ''A'' in ''S''.