Quantum Probability - Mathematical Definition

Mathematical Definition

The basic definition in quantum probability is that of a quantum probability space, sometimes also referred to as an algebraic or noncommutative probability space.

Definition : Quantum probability space.

A pair (A, P), where A is a *-algebra and P is a state, is called a quantum probability space.

This definition is a generalization of the definition of a probability space in Kolmogorovian probability theory, in the sense that every (classical) probability space gives rise to a quantum probability space if A is chosen as the *-algebra of bounded complex-valued measurable functions on it.

The projections pA are the events in A, and P(p) gives the probability of the event p.

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