As mentioned previously, a classical Markov semigroup of transition operators { T t , t ≥ 0} can be generated from a classical Markov process { X t , t ≥ 0}. On the other hand, given a Markov semigroup, a Markov process { X t , t ≥ 0} can be constructed based on the Markov semigroup { T t , t ≥ 0} using the Kolmogorov consistency theorem. In the context of quantum probability space ( A , ρ ), a Markov dilation (i.e., construction of a quantum Markov process from a given quantum Markov semigroup of operators { I t , t ≥ 0}) turns out to be a nontrivial matter. This is partly due to the fact that, although the general concept of conditional expectation of an observable given a sub-von Neumann algebras B ⊂ A can be defined and required properties can be described, an explicit construction of such a conditional expectation is still unavailable in general. The main objective of this chapter is to introduce relevant concepts and to develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. There are 2 major components in this chapter: (i) introduction of concepts and derivation of properties of a quantum Markov processes based on some assumed and/or derived properties of conditional expectation E ρ [· | A t ] ] given a filtration of sub-von Neumann algebras { A t ] , t ≥ 0} of A , and (ii) Markov dilation or construction of a weak quantum Markov flow (WQMF) from a given quantum Markov semigroup { I t , t ≥ 0} using the weak conditional expectation E [· | F t ], where { F t , t ≥ 0} is a filtration of orthogonal projection operators defined on the complex Hilbert space ℍ. The main objective of the first component is to introduce the relevant concept and develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. Although the theory of (classical) Markov processes is well known in the probability community (see, for example, Dynkin [Dyn62], Ethier and Kurtz [EK85]), we outline some basic concept and results from the classical theory of homogeneous Markov processes below for comparison purpose.
As mentioned previously, a classical Markov semigroup of transition operators { T t , t ≥ 0} can be generated from a classical Markov process { X t , t ≥ 0}. On the other hand, given a Markov semigroup, a Markov process { X t , t ≥ 0} can be constructed based on the Markov semigroup { T t , t ≥ 0} using the Kolmogorov consistency theorem. In the context of quantum probability space ( A , ρ ), a Markov dilation (i.e., construction of a quantum Markov process from a given quantum Markov semigroup of operators { I t , t ≥ 0}) turns out to be a nontrivial matter. This is partly due to the fact that, although the general concept of conditional expectation of an observable given a sub-von Neumann algebras B ⊂ A can be defined and required properties can be described, an explicit construction of such a conditional expectation is still unavailable in general. The main objective of this chapter is to introduce relevant concepts and to develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. There are 2 major components in this chapter: (i) introduction of concepts and derivation of properties of a quantum Markov processes based on some assumed and/or derived properties of conditional expectation E ρ [· | A t ] ] given a filtration of sub-von Neumann algebras { A t ] , t ≥ 0} of A , and (ii) Markov dilation or construction of a weak quantum Markov flow (WQMF) from a given quantum Markov semigroup { I t , t ≥ 0} using the weak conditional expectation E [· | F t ], where { F t , t ≥ 0} is a filtration of orthogonal projection operators defined on the complex Hilbert space ℍ. The main objective of the first component is to introduce the relevant concept and develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. Although the theory of (classical) Markov processes is well known in the probability community (see, for example, Dynkin [Dyn62], Ethier and Kurtz [EK85]), we outline some basic concept and results from the classical theory of homogeneous Markov processes below for comparison purpose.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).
It is widely known that the commutative probability theory based on Kolmogorov's measure theoretic axioms and the quantum probability theory based on von Neumann's postulates were both created at about the same time in the 1930s (see, e.g., Kolmogorov [Kol50] and von Neumann [vNeu55] for the origins of these 2 very different but parallel lines of theory). Following Kolmogorov's original work, theory of the classical Markov processes has been the subject of intensive research for the last few decades. The classical theory, its applications, and their connections to other areas of research have been systematically developed by many prominent probabilists as illuminated by their widely referenced monographs such as those of Dynkin [Dyn62], Blumenthal and Getoor [BG68], Ethier and Kurtz [EK85], Renuv and Yor [RY99], and others. On the other hand, development of a complete theory of quantum stochastics such as quantum Markov processes has been progressing at a much slower pace in comparison with that of its classical counterpart. This is perhaps because development of a theory of quantum stochastics and quantum Markov processes requires an unusually large number of tools from operator theory and perhaps also because the probabilistic and analytical tools for understanding sample path behaviors of quantum stochastic processes have yet to be developed. The main goal of this monograph is to give a systematic exploration of relevant topics in quantum Markov processes in hopes of stimulating further research along this line and of stimulating interest in the classical stochastics community for research in its quantum counterpart. This monograph is written largely based on the current account of relevant research results by widely surveying relevant results contributed by many prominent researchers in quantum probability, quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups, strong quantum Markov processes, and large time asymptotic behaviors of quantum Markov semigroups through a systematic and self-contained introduction/presentation of these very interesting topics in an attempt to illuminate the rigor and beauty of quantum stochastics and quantum Markov processes.
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 Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ , P ) is replaced by the pair ( L ∞ (Ω), E (.) = ∫ Ω . d P ) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space ( N ,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form ( L ∞ (Ω), E ), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S ( H )), if we take N = B ( H ) and Ф the state defined by Ф( X ) = 〈 ψ, X ψ 〉 (or Ф ( X ) = tr(ρ X )) for X ∊ B ( H ). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces. A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments. However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.
We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.