Using Classical Probability To Guarantee Properties of Infinite Quantum Sequences

Abstract

We consider the product of infinitely many copies of a spin-$\frac{1}{2}$ system. We construct projection operators on the corresponding nonseparable Hilbert space which measure whether the outcome of an infinite sequence of ${\sigma }^x$ measurements has any specified property. In many cases, product states are eigenstates of the projections, and therefore the result of measuring the property is determined. Thus we obtain a nonprobabilistic quantum analogue to the law of large numbers, the randomness property, and all other familiar almost-sure theorems of classical probability.

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