Indeterminacy For Mixed States
We have described indeterminacy for a quantum system which is in a pure state. Mixed states 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
for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:
This is a probability measure defined on the Borel subsets of R which is the probability distribution obtained by measuring A in S.
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