Introduction to Quantum Stochastic Calculus

• I Events, Observables and States.- 1 From classical to quantum probability.- 2 Notational preliminaries.- 3 Finite dimensional quantum probability spaces.- 4 Observables in a simple quantum probability space.- 5 Variance and covariance.- 6 Dynamics in finite dimensional quantum probability spaces.- 7 Observables with infinite number of values and the Hahn-Hellinger Theorem.- 8 Probability distributions on? (?)and Gleason’s Theorem.- 9 Trace class operators and Schatten’s Theorem.- 10 Spectral integration and Stone’s Theorem on the unitary representations of ?pk.- 11 Basic notions of the theory of unbounded operators.- 12 Spectral integration of unbounded functions and von Neumann‘s Spectral Theorem.- 13 Stone generators, characteristic functions and moments.- 14 Wigner’s Theorem on the automorphisms of ?(?).- II Observables and States in Tensor Products of Hubert Spaces.- 15 Positive definite kernels and tensor products of Hilbert Spaces.- 16 Operators in tensor products of Hilbert Spaces.- 17 Symmetric and antisymmetric tensor products.- 18 Examples of discrete time quantum stochastic flows.- 19 The Fock Spaces.- 20 The Weyl Representation.- 21 Weyl Representation and infinitely divisible distributions.- 22 The symplectic group of ? and Shale’s Theorem.- 23 Creation, conservation and annihilation operators in ?a(?).- III Stochastic Integration and Quantum Ito’s Formula.- 24 Adapted processes.- 25 Stochastic integration with respect to creation, conservation and annihilation processes.- 26 A class of quantum stochastic differential equations.- 27 Stochastic differential equations with infinite degrees of freedom.- 28 Evans-Hudson Flows.- 29 A digression on completely positive linear maps and Stinespring’s Theorem.- 30 Generators of quantumdynamical semigroups and the Gorini, Kossakowski, Sudarshan, Lindblad Theorem.- References.- Author Index.

"Elegantly written, with obvious appreciation for fine points of higher mathematics...most notable is [the] author's effort to weave classical probability theory into [a] quantum framework." --The American Mathematical Monthly "This is an excellent volume which will be a valuable companion both for those who are already active in the field and those who are new to it. Furthermore there are a large number of stimulating exercises scattered through the text which will be invaluable to students." --Mathematical Reviews