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Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes

Anilesh Mohari, Kalyan B. Sinha

Published Jan 1, 1990• 26 citations
OpenAlex

Abstract

Quantum stochastic flows are constructed for infinite degrees of freedom. The theory is then used to show that a classical countable state Markov process can be looked upon as one such commutative stochastic flow.

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This paper consists of 333 parts. The first part only considers classical processes and introduces two different extensions of the notion of hidden Markov process. In the second part, the notion of quantum hidden process is introduced. In the third part it is proven that, by restricting various types of quantum Markov chains to appropriate commutative sub-algebras (diagonal sub-algebras) one recovers all the classical hidden process and, in addition, one obtains families of processes which are not usual hidden Markov process, but are included in the above mentioned extensions of these processes. In this paper we only deal with processes with an at most countable state space.
A framework for finite-dimensional quantum Markov chains on Hilbert spaces is introduced. Quantum Markov chains generalize both classical Markov chains with possibly hidden states and existing models of quantum walks on finite graphs. Quantum Markov chains are based on Markov operations that may be applied to quantum systems and include quantum measurements, for example. It is proved that quantum Markov chains are asymptotically stationary and hence possess ergodic and entropic properties. With a quantum Markov chain one may associate a quantum Markov process, which is a stochastic process in the classical sense. Generalized Markov chains allow a representation with respect to a generalized Markov source model with definite (but possibly hidden) states relative to which observables give rise to classical stochastic processes. It is demonstrated that this model allows for observables to violate Bell’s inequality.
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.
Random quantum processes with a countable set of elementary orthogonal events that govern the operation of a quantum computer are discussed. For the case of pure states, expressions for a posteriori quantum probabilities are derived. For stationary and nonstationary random quantum processes with the countable set of states, properties of transition quantum probabilities are investigated. A method for finding a posteriori quantum probabilities is proposed. An analog of the Kolmogorov equation for random quantum processes is presented.
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.