Quantum Indeterminacy - Indeterminacy For Mixed States

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.