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**An introduction to quantum stochastic calculus.**
*(English)*
Zbl 0751.60046

Monographs in Mathematics. 85. Basel etc.: Birkhäuser. xi, 290 p. (1992).

Quantum stochastic calculus is a new field on borders of stochastic processes, operator theory and quantum theory of open systems. Mathematically, it emerges on the intersection of the two fundamental concepts — second quantization and linearly ordered time filteration, and has its roots in the continuous tensor product structure of the underlying Fock space, studied since the 60’s in the works of R. F. Streater, H. Araki, K. R. Parthasarathy and K. Schmidt, A. Guichardet and others. Quantum stochastic calculus provides new and often unexpected links between classical and quantum stochastics, leading in particular to a unification of continuous and jump processes; reveals an algebraic background of the stochastic integral and of the Itô’s formula; allows stochastic representations for the dynamics of open quantum systems through solutions of quantum stochastic differential equations. This monograph gives a systematic and self-contained introduction to the Fock space quantum stochastic calculus in its basic form, introduced by R. L. Hudson and the author in 1984.

The monograph is divided into three clear-cut parts. Chapter I “Events, observables and states” gives a necessary background in operator theory and noncommutative probability. The central concept is that of quantum observable viewed from three different points: as a spectral measure on the real line, as a selfadjoint operator in a Hilbert space, and as a unitary representation of the additive group of the real line. A number of classical operator-theoretic results, such as Hahn-Hellinger’s, Stone’s, von Neumann’s, Schatten’s theorems and also Gleason’s and Wigner’s theorems is presented with complete proofs, some of which are modified and adapted by the author. Special attention is paid to concrete examples of quantum observables and their distributions.

Chapter II “Observables and states in tensor products of Hilbert spaces” provides an account of second quantization adapted to the needs of quantum stochastic calculus. It begins with a description of Hilbert tensor products based on properties of positive definite kernels. Discrete quantum flows are described as noncommutative analogs of Markov chains, with many interesting examples. Embedding of infinitely divisible random variables in the Fock space and Shale’s theorem on unitary implementation of symplectic automorphisms are considered in detail. Along with basic material, recent developments, such as free stochastic calculus or kernel algebra are discussed or indicated in numerous examples, exercises and in notes.

The final goal is achieved in Chapter III “Stochastic integration and quantum Itô’s formula”. Here the original construction of the quantum stochastic integral with respect to creation, conservation and annihilation processes in the Fock space is presented, resulting in the quantum Itô’s product formula; an existence theorem for solutions of quantum stochastic differential equations with infinite number of degrees of freedom is proved, giving rise to quantum diffusions — the Evans- Hudson flows; a stochastic representation of a quantum dynamical semigroup through the flow is exhibited. The book ends with a digression on completely positive maps and Stinespring’s theorem and with the proof of the Lindblad-Gorini-Kossakowski-Sudarshan representation for the generator of a norm-continuous quantum dynamical semigroup, complemented by the study of uniqueness of the representation.

The field of the noncommutative probability has undergone fast development in the past two decades and certainly deserves broader recognition, in particular, among probability theorists. However, there are serious obstacles preventing from quick recognition: one is relative conceptual novelty and complexity of qantum phenomenology, requiring from a novice intellectual efforts comparable with transition from determinism to classical probability; the other is a necessity of using the mathematics of operators rather different or even disjoint from the familiar measure-theoretic background. In Parthasarathy’s book these difficulties are managed by making emphasis on the mathematical aspects of quantum formalism and its connections with classical probability and by extensive presentation of carefully selected functional analytic material. This makes the book very convenient for a reader with the probability-theoretic orientation, wishing to make acquaintance with wonders of the noncommutative probability, and, more specifically, for a mathematics student studying this field.

The monograph is divided into three clear-cut parts. Chapter I “Events, observables and states” gives a necessary background in operator theory and noncommutative probability. The central concept is that of quantum observable viewed from three different points: as a spectral measure on the real line, as a selfadjoint operator in a Hilbert space, and as a unitary representation of the additive group of the real line. A number of classical operator-theoretic results, such as Hahn-Hellinger’s, Stone’s, von Neumann’s, Schatten’s theorems and also Gleason’s and Wigner’s theorems is presented with complete proofs, some of which are modified and adapted by the author. Special attention is paid to concrete examples of quantum observables and their distributions.

Chapter II “Observables and states in tensor products of Hilbert spaces” provides an account of second quantization adapted to the needs of quantum stochastic calculus. It begins with a description of Hilbert tensor products based on properties of positive definite kernels. Discrete quantum flows are described as noncommutative analogs of Markov chains, with many interesting examples. Embedding of infinitely divisible random variables in the Fock space and Shale’s theorem on unitary implementation of symplectic automorphisms are considered in detail. Along with basic material, recent developments, such as free stochastic calculus or kernel algebra are discussed or indicated in numerous examples, exercises and in notes.

The final goal is achieved in Chapter III “Stochastic integration and quantum Itô’s formula”. Here the original construction of the quantum stochastic integral with respect to creation, conservation and annihilation processes in the Fock space is presented, resulting in the quantum Itô’s product formula; an existence theorem for solutions of quantum stochastic differential equations with infinite number of degrees of freedom is proved, giving rise to quantum diffusions — the Evans- Hudson flows; a stochastic representation of a quantum dynamical semigroup through the flow is exhibited. The book ends with a digression on completely positive maps and Stinespring’s theorem and with the proof of the Lindblad-Gorini-Kossakowski-Sudarshan representation for the generator of a norm-continuous quantum dynamical semigroup, complemented by the study of uniqueness of the representation.

The field of the noncommutative probability has undergone fast development in the past two decades and certainly deserves broader recognition, in particular, among probability theorists. However, there are serious obstacles preventing from quick recognition: one is relative conceptual novelty and complexity of qantum phenomenology, requiring from a novice intellectual efforts comparable with transition from determinism to classical probability; the other is a necessity of using the mathematics of operators rather different or even disjoint from the familiar measure-theoretic background. In Parthasarathy’s book these difficulties are managed by making emphasis on the mathematical aspects of quantum formalism and its connections with classical probability and by extensive presentation of carefully selected functional analytic material. This makes the book very convenient for a reader with the probability-theoretic orientation, wishing to make acquaintance with wonders of the noncommutative probability, and, more specifically, for a mathematics student studying this field.

Reviewer: A.S.Holevo (Moskva)

### MSC:

60Hxx | Stochastic analysis |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

81S25 | Quantum stochastic calculus |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |