Quantum state diffusion.

*(English)*Zbl 0946.60093
Cambridge: Cambridge University Press. xi, 176 p. (1998).

This book addresses to scientists interested in open quantum systems and the foundations of quantum mechanics. As the author stresses himself, the face of quantum physics has been changed by recent experiments with detailed control over individual quantum systems. These systems include atoms at low temperature, entangled photons in cavities as well as quantum systems used in new and future technologies such as quantum cryptography and quantum computation. According to the author these experiments need new theories. The author’s response to this challenge is based upon quantum state diffusions, a field whose systematic approach is presented here for the first time.

The book is organised as follows. Starting with classical Brownian motion and Itô calculus, it goes through open quantum systems in the subsequent chapter. These two chapters prepare the reader for the delightful main course of the book: the definition and main properties of quantum state diffusion, assorted with a subsequent chapter on localisation. After that, numerical methods and examples are presented. The discussion on quantum foundations, as well as an insight on primary state diffusion follows then. The book ends with a chapter on the classical dynamics of quantum localisation and a chapter on the semiclassical theory of ensembles and linear dynamics.

This is a highly motivating and well-written book. Historical references are spread all over the text taking the reader along the endless debate on the foundations of quantum theory. The author has accomplished an admirable pedagogical effort. Each chapter starts with a small table of contents followed by a brief summary. The style is direct, avoiding superfluous technicalities. It is a good introduction for physicists to methods and concepts of classical stochastic analysis, although the mathematical tools employed correspond to the state of that theory during the seventies. Indeed, open quantum systems have motivated other mathematical approaches. Namely, the emergent theory of quantum stochastic differential equations provides an extension of classical stochastic processes, which is particularly well adapted to quantum physics.

The author includes within his references the book of K. R. Parthasarathy [“An introduction to quantum stochastic calculus” (1992; Zbl 0751.60046)]. However, no explicit discussion is developed on the relations between quantum state diffusions and quantum Markov flows obtained as solutions to stochastic differential equations. Quantum mechanics, and a fortiori, open quantum systems, contain implicitly a model of randomness. The role of quantum stochastic analysis has been to extract mathematical models for stochastic processes from quantum physics. Namely, the essence of this new stochastic calculus resides in the existence of operator algebras where both, the canonical commutation relations (CCR) and non-commutative Itô tables for multiplication of differentials, have a rigorous interpretation [see for instance R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058), P.-A. Meyer, “Quantum probability for probabilists” (1993; Zbl 0773.60098), and K. R. Parthasarathy (loc. cit.)]. Thus, combinations of creation, annihilation or gauge operators, depending on a time parameter, extend classical Brownian motion or Poisson processes. In the above framework the key notions are those of quantum Markov flow (QMF) and quantum dynamical semigroup (QDS). Introduced by physicists during the seventies, these semigroups (QDS) were aimed at providing a suitable formalism for studying the evolution of open systems. An open quantum system involves dissipative effects modelled through the mutual interaction of different subsystems. Usually, the tensor product of a given initial Hilbert space, representing the free evolution with a Fock space, used to describe different sources of interaction, gives the space of the whole evolution. The QMF of the total system satisfies a quantum stochastic differential equation with quantum noises like creation, annihilation or gauge operators defined on the space of the total evolution [see P.-A. Meyer (loc. cit), K. R. Parthasarathy (loc. cit.) and F. Fagnola, Probab. Theory Relat. Fields 86, No. 4, 501-516 (1990; Zbl 0685.60058)]. The corresponding QDS is obtained by projecting the action of the QMF to the initial space.

The description of the evolution of an important class of open systems was achieved in the seventies by giving the explicit form of the Liouvillian map (or semigroup generator) ruling the dynamics. This was done, in the strongly continuous case, by V. Gorini, A. Kossakowski and E. C. G. Sudarshan [J. Math. Phys. 17, 821-825 (1976)], G. Lindblad [Commun. Math. Phys. 48, 119-130 (1976; Zbl 0343.47031)] and further extended by E. B. Davies [J. Funct. Anal. 34, 421-432 (1979; Zbl 0428.47021)]. At present, the generators of QDS are referred as the Lindblad map or the Lindbladian. The author makes strong use of the Lindblad map as well. Indeed, there are different ways leading to a quantum dynamical semigroup since it represents the projection over the initial space of the evolution of a system defined in a bigger space (a dilation of the QDS). Thus, quantum state diffusion dilates a given QDS too, but that dilation uses classical stochastic noises instead of non-commutative noises, which are inherent to the quantum theory. From this point of view, quantum state diffusions should be considered as a simulation of a QDS (or of a QMF) through classical stochastic processes. This makes particularly interesting the numerical aspects of this approach. On the other hand, the approach via QMF has been successful in building up models for quantum optics, for instance see F. Fagnola, R. Rebolledo and C. Saavedra [J. Math. Phys. 35, No. 1, 1-12 (1994; Zbl 0797.60052) and in: Proceedings of the 2nd international workshop on stochastic analysis and mathematical physics, 61-71 (1988; Zbl 0940.81052)] and F. Fagnola and R. Rebolledo [in: Proc. Univ. Udine Conf. in honour of A. Frigerio, 73-86 (1996)] and other concrete examples of open quantum systems. However, the most valuable theoretical contributions of this kind of approach reside on the qualitative analysis of quantum dynamical systems. This includes ergodic theorems, the criteria on the existence of stationary states and the convergence towards the equilibrium, see F. Fagnola and R. Rebolledo [C. R. Acad. Sci., Paris, Sér. I 321, No. 4, 473-476 (1995; Zbl 0842.58040) and Infin. Dimens. Anal. Quantum Probab. Relat. Top 1, No. 4, 561-572 (1998; Zbl 0923.46073)]. Moreover, Fagnola has studied quantum diffusions as a non-commutative dilation of classical diffusion processes.

To summarise, the book provides physicists with an appealing introduction to methods and concepts of stochastic analysis. Moreover, it illustrates a way of implementing specific numerical procedures for open quantum systems. That feature will certainly interest both physicists and mathematicians who will enjoy as well the philosophical discussion on the foundations of quantum mechanics. This book, as any other scientific book, is itself an open system which the interested reader should complement by consulting the current research books on quantum stochastic analysis like that of Meyer (loc. cit.) and Parthasarathy (loc. cit.), as well as the ever increasing flow of publications within this emergent field.

The book is organised as follows. Starting with classical Brownian motion and Itô calculus, it goes through open quantum systems in the subsequent chapter. These two chapters prepare the reader for the delightful main course of the book: the definition and main properties of quantum state diffusion, assorted with a subsequent chapter on localisation. After that, numerical methods and examples are presented. The discussion on quantum foundations, as well as an insight on primary state diffusion follows then. The book ends with a chapter on the classical dynamics of quantum localisation and a chapter on the semiclassical theory of ensembles and linear dynamics.

This is a highly motivating and well-written book. Historical references are spread all over the text taking the reader along the endless debate on the foundations of quantum theory. The author has accomplished an admirable pedagogical effort. Each chapter starts with a small table of contents followed by a brief summary. The style is direct, avoiding superfluous technicalities. It is a good introduction for physicists to methods and concepts of classical stochastic analysis, although the mathematical tools employed correspond to the state of that theory during the seventies. Indeed, open quantum systems have motivated other mathematical approaches. Namely, the emergent theory of quantum stochastic differential equations provides an extension of classical stochastic processes, which is particularly well adapted to quantum physics.

The author includes within his references the book of K. R. Parthasarathy [“An introduction to quantum stochastic calculus” (1992; Zbl 0751.60046)]. However, no explicit discussion is developed on the relations between quantum state diffusions and quantum Markov flows obtained as solutions to stochastic differential equations. Quantum mechanics, and a fortiori, open quantum systems, contain implicitly a model of randomness. The role of quantum stochastic analysis has been to extract mathematical models for stochastic processes from quantum physics. Namely, the essence of this new stochastic calculus resides in the existence of operator algebras where both, the canonical commutation relations (CCR) and non-commutative Itô tables for multiplication of differentials, have a rigorous interpretation [see for instance R. L. Hudson and K. R. Parthasarathy, Commun. Math. Phys. 93, 301-323 (1984; Zbl 0546.60058), P.-A. Meyer, “Quantum probability for probabilists” (1993; Zbl 0773.60098), and K. R. Parthasarathy (loc. cit.)]. Thus, combinations of creation, annihilation or gauge operators, depending on a time parameter, extend classical Brownian motion or Poisson processes. In the above framework the key notions are those of quantum Markov flow (QMF) and quantum dynamical semigroup (QDS). Introduced by physicists during the seventies, these semigroups (QDS) were aimed at providing a suitable formalism for studying the evolution of open systems. An open quantum system involves dissipative effects modelled through the mutual interaction of different subsystems. Usually, the tensor product of a given initial Hilbert space, representing the free evolution with a Fock space, used to describe different sources of interaction, gives the space of the whole evolution. The QMF of the total system satisfies a quantum stochastic differential equation with quantum noises like creation, annihilation or gauge operators defined on the space of the total evolution [see P.-A. Meyer (loc. cit), K. R. Parthasarathy (loc. cit.) and F. Fagnola, Probab. Theory Relat. Fields 86, No. 4, 501-516 (1990; Zbl 0685.60058)]. The corresponding QDS is obtained by projecting the action of the QMF to the initial space.

The description of the evolution of an important class of open systems was achieved in the seventies by giving the explicit form of the Liouvillian map (or semigroup generator) ruling the dynamics. This was done, in the strongly continuous case, by V. Gorini, A. Kossakowski and E. C. G. Sudarshan [J. Math. Phys. 17, 821-825 (1976)], G. Lindblad [Commun. Math. Phys. 48, 119-130 (1976; Zbl 0343.47031)] and further extended by E. B. Davies [J. Funct. Anal. 34, 421-432 (1979; Zbl 0428.47021)]. At present, the generators of QDS are referred as the Lindblad map or the Lindbladian. The author makes strong use of the Lindblad map as well. Indeed, there are different ways leading to a quantum dynamical semigroup since it represents the projection over the initial space of the evolution of a system defined in a bigger space (a dilation of the QDS). Thus, quantum state diffusion dilates a given QDS too, but that dilation uses classical stochastic noises instead of non-commutative noises, which are inherent to the quantum theory. From this point of view, quantum state diffusions should be considered as a simulation of a QDS (or of a QMF) through classical stochastic processes. This makes particularly interesting the numerical aspects of this approach. On the other hand, the approach via QMF has been successful in building up models for quantum optics, for instance see F. Fagnola, R. Rebolledo and C. Saavedra [J. Math. Phys. 35, No. 1, 1-12 (1994; Zbl 0797.60052) and in: Proceedings of the 2nd international workshop on stochastic analysis and mathematical physics, 61-71 (1988; Zbl 0940.81052)] and F. Fagnola and R. Rebolledo [in: Proc. Univ. Udine Conf. in honour of A. Frigerio, 73-86 (1996)] and other concrete examples of open quantum systems. However, the most valuable theoretical contributions of this kind of approach reside on the qualitative analysis of quantum dynamical systems. This includes ergodic theorems, the criteria on the existence of stationary states and the convergence towards the equilibrium, see F. Fagnola and R. Rebolledo [C. R. Acad. Sci., Paris, Sér. I 321, No. 4, 473-476 (1995; Zbl 0842.58040) and Infin. Dimens. Anal. Quantum Probab. Relat. Top 1, No. 4, 561-572 (1998; Zbl 0923.46073)]. Moreover, Fagnola has studied quantum diffusions as a non-commutative dilation of classical diffusion processes.

To summarise, the book provides physicists with an appealing introduction to methods and concepts of stochastic analysis. Moreover, it illustrates a way of implementing specific numerical procedures for open quantum systems. That feature will certainly interest both physicists and mathematicians who will enjoy as well the philosophical discussion on the foundations of quantum mechanics. This book, as any other scientific book, is itself an open system which the interested reader should complement by consulting the current research books on quantum stochastic analysis like that of Meyer (loc. cit.) and Parthasarathy (loc. cit.), as well as the ever increasing flow of publications within this emergent field.