# Quantum Stochastics: 37 (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 37)

Cambridge University Press, 2/19/2015

EAN 9781107069190, ISBN10: 110706919X

Hardcover, 424 pages, 26.4 x 18.5 x 3 cm

Language: English

Originally published in English

The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.

Introduction and summary

1. Operator algebras and topologies

2. Quantum probability

3. Quantum stochastic calculus

4. Quantum stochastic differential equations

5. Quantum Markov semigroups

6. Minimal QDS

7. Quantum Markov processes

8. Strong quantum Markov processes

9. Invariant normal states

10. Recurrence and transience

11. Ergodic theory.