Quantum probability
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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.
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[edit] 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.
[edit] Motivation
In classical probability theory, information is summarized by sigma-algebra A ∈ F of events in a classical probability space (Ω ,F,P). For example, A could be the σ-algebra σ(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 ∈ A implies A ∈ A .
A state P on A is a linear map P : A → C such that 0≤P(A* A) for all A \in; A (positivity) and P(I) = 1 (normalization). A projection is an operator P ∈ A such that P2 = P = P .
[edit] 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 P ∈ A are the events in A , and P(P) gives the probability of the event P.
[edit] References
- L. Accardi, A. Frigerio, and J.T. Lewis, Quantum stochastic processes, Publ. Res. Inst. Math. Sci. 18 (1982), no. 1, 97--133.
- V. P. Belavkin,Optimal linear randomized filtration of quantum boson signals, Problems of Control and Information Theory, 3:1, pp. 47-62, 1972/1974.
- V. P. Belavkin, Optimal multiple quantum statistical hypothesis testing, Stochastics, Vol. 1, pp. 315-345, Gordon & Breach Sci. Pub., 1975.
- V. P. Belavkin, Optimal quantum filtration of Makovian signals [In Russian], Problems of Control and Information Theory, 7:5, pp. 345-360, 1978.
- V. P. Belavkin, Theory of the control of observable quantum systems, Autom. Rem. Control, 44 (1983), pp. 178-188.
- R.L. Hudson, K.R. Parthasarathy, "Quantum Ito's formula and stochastic evolutions", Comm. Math. Phys. 93 (1984), no. 3, 301--323.
- P.-A. Meyer, Quantum probability for probabilists, Lecture Notes in Mathematics Vol 1538, Springer-Verlag, Berlin, 1993.
- K.R. Parthasarathy, "An introduction to quantum stochastic calculus", Monographs in Mathematics, 85, Birkhäuser Verlag, Basel, 1992
- D. Voiculescu, K. Dykema, A. Nica, "Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups," CRM Monograph Series vol. 1, American Mathematical Society, Providence, RI, 1992.
- John von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Mathematische Annalen, volume 102, pages 49-131, 1929.
- John von Neumann, Mathematische Grundlagen der Quantenmechanik, (Die Grundlehren der Mathematischen Wissenschaften, Band 38.) Berlin, Springer, 1932.
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