Quantum probability

Quantum probability was developed in the 1980's as a noncommutative analog of the Kolmogorovian stochastic  processes theory. One of its aims is to clarify the probabilistic mathematical foundations of quantum theory and its statistical interpretation.

Significant recent applications to physics include the dynamical solution of quantum measurement problem by giving constructive models of quantum observation processes resolving many famous paradoxes of quantum mechanics.

The most recent advances are based on quantum stochastic filtering and feedback control theory as application of quantum stochastic calculus method.

Orthodox Quantum Mechanics
Orthodox Quantum Mechanics has two seemingly contradictory mathematical descriptions:

1. Deterministic (read unitary evolution / Schrödinger equation); and a

2. stochastic (random) wavefunction collapse.

Most physicists are not concerned with this apparent problem. Physical intuition usually provides the answer, and only in unphysical systems (Schrödinger's cat, isolated atom) do paradoxes seem to occur.

Orthodox Quantum mechanics can be reformulated in a Quantum Probability (80's, 90's) framework, where Quantum Filtering (2005 (or Belavkin, 1970s)) is the natural description of the process of measurement. This new framework encapsulates the standard postulates of quantum mechanics, and thus all the science involved in the orthodox postualtes. However, it becomes more beautiful by removing the "mathematical contradictions" we ordinarily see.

Motivation
In classical probability theory, information is summarized by sigma-algebra A &isin; F of events in a classical probability space (&Omega; ,F,P). For example, A could be the &sigma;-algebra &sigma;(X) generated by a random variable X, which contains information on the values taken by X. We wish to describe quantum information in similar algebraic terms in such a way to capture the non-commutative features and the information made available in an experiment. The appropriate algebraic structure for observables, or more generally, operators, is the *-algebra. A (unital) *- algebra is a complex vector subspace A of operators on a Hilbert space H that
 * contains the identity I and
 * is closed under composition (a multiplication) and adjoints (an involution): A &isin; A implies A† &isin; A.

A state P on A is a linear map P : A &rarr; C such that 0≤P(A* A) for all A \in; A (positivity) and P(I) = 1 (normalization). A projection is an operator P &isin; A such that P2 = P = P†.

Mathematical definition
One mathematical definition is provided by L.Bouten, R. Van Handel and M.James, Introduction to quantum filtering for a finite dimensional system (1)


 * Definition : Finite dimensional quantum probability space.

A pair (N, P), where N is a *-algebra and P is a state, is called a finite dimensional quantum probability space. The projections P &isin; N are the events in N, and P(P) gives the probability of the event P.

A similar definition for an infinite dimensional quantum probability space exists.