Finite Quantum Measure Spaces

Measure and integration theory is a well established field of mathematics that is over a hundred years old. The theory possesses many deep and elegant theorems and has important applications in functional analysis, probability theory and theoretical physics. Measure theory can be applied whenever you are measuring something whether it be length, volume, probabilities, mass, energy, etc. Although finite measure theory, in which the measure space has only a finite number of elements, is much simpler than the general theory, it also has important applications to probability theory, combinatorics and computer science. In this article we shall discuss a generalization called finite quantum measure spaces. Just as quantum mechanics possesses a certain “quantum weirdness,” these spaces lack some of the simplicity and intuitive nature of their classical counterparts. Although there is a general theory of quantum measure spaces, we shall only consider finite spaces to keep technicalities to a minimum. Nevertheless, these finite spaces still convey the flavor of the subject and exhibit some of the unusual properties of quantum objects. Much of this unusual behavior is due to a phenomenon called quantum interference which is a recurrent theme in the present article.