QUANTUM PROBABILITY AND STOCHASTICS

Quantum Probability Theory (QPT)
is not a particular kind of applied classical probability. Rather opposite: The whole Kolmogorov probability theory is a special, or limit case of the new (quantum) probability theory, in the same sense as classical mechanics is a special (limit) case of quantum mechanics. Since the time when Dirac and von Neumann understood the observables in quantum mechanics as non-commutative entries generalizing the commutative real random variables of classical probability, it has been clear that the probabilities of quantum occurrences may arise not just as a result of incomplete knowledge of the quantum state. QPT is a mathematical framework underpinning the calculation of probabilities for noncommutative observables represented by operators on a Hilbert space. QPT is also known under the name of Noncommutative Probability Theory, and Free Probability in the special, free case. More on the development of QPT in the Quantum Century one can read in my recent review paper [1], and on logico-theoretical foundations of quantum probability and related quantum structures - in [2, 6, 7].

Quantum Stochastic Process (QSP)
is now the main object of study in QPT starting from the beginning of 80's. It is defined as a family of not necessarily commuting quantum observables, or more general, as a field of operator algebras indexed by a causal set and represented on the same Hilbert space with a fixed quantum state. Just as the classical stochastic process is represented by a family of commuting random variables on the same probability space. This notion was introduced independently by Accardi, Frigerio and Lewis, and by myself [3] in a more general, field setup. The noncommutative generalization [4, 5] of the Main Kolmogorov Theorem for reconstruction of classical stochastic processes in a projective limit uniquely defines the QSP with an increasing identity on the minimal filtering Hilbert space. My canonical construction of the weak QSP and the corresponding uniqueness theorem was rediscovered almost 10 years later by Bhat and Parthasarathy for the one-parameter case of weak quantum Markov process. The appearance of quantum posterior stochastics in QPT is explained in [8], and interplay of classical and quantum stochastics for quantum diffusion, measurement and filtering is described in [9,10].

My 10 Relevant Publications:
  1. V. P. Belavkin: Quantum Probabilities and Paradoxes of the Quantum Century. Infinite Dim. Analysis, Quantum Prob. & Related Topics 3 (4) 577-610 (2000). math.PR/0512415, PDF.
  2. V. P. Belavkin: Semilogics, Quasilogics and Other Quantum Structures. Tatra Mout. Math. Publ. 10 199--223 (1997). math.PR/0512413, PDF.
  3. V. P. Belavkin: Reconstruction Theorem for Quantum Stochastic Fields. Usp Math Nauk (Russian Math Surveys) 2 137--138 (1984).
  4. V. P. Belavkin: Reconstruction Theorem for Quantum Stochastic Processes. Theoretical and Mathematical Physics 3 409--431 (1985). math.PR/0512410, PDF.
  5. V. P. Belavkin: A Non-Commuting Analog of the Main Kolmogorov Theorem. In Proc of Fourth International Conference on Probability Theory and Mathematical Statistics 2 45--150. Vilnius, 1985.
  6. V. P. Belavkin: Ordered *- Semirings and Generating Functionals of Quantum Statistics. Soviet Math. Dokl. 35 (2) 246--249 (1987).
  7. V. P. Belavkin: Mathematical Foundation of General Systems Theory. Textbook Moscow Institute Electronics and Mathematics, Moscow 1987.
  8. V. P. Belavkin: A Quantum Posterior Stochastics and Spontaneous Collapse. In Stochastics and Quantum Mechanics 40--68. World Scientific, Singapore 1990.
  9. V. P. Belavkin: Quantum Diffustion, their Measurement and Filtering. Probability Theory and its Application 38, 39 (4) 742--757, 640--658 (1993, 1994). quant-ph/0510028, PDF.
  10. V. P. Belavkin: The Interplay of Classical and Quantum Stochastics: Diffusion, Measurement and Filtering. In Chaos -- The Interplay Between Stochastic and Deterministic Behaviour 21--41. Lecture Notes in Physics, Springer, Berlin 1995.