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Radiophysics and Quantum Electronics

Conjugation conditions for the transition probability density of a multidimensional Markov process

M. L. Tai

Published Jan 1, 1966• 0 citations
Publisher
DOI
10.1007/bf01038344

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Quantum Markov Processes

Mou-Hsiung Chang

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.

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Hidden stochastic, quantum and dynamic information of Markov diffusion process and its evaluation by an entropy integral measure under the impulse controls actions, applied to information observer

Vladimir S. Lerner

Hidden information emerges under impulse interactions with Markov diffusion process modeling interactive random environment. Impulse yes no action cuts Markov correlations revealing Bit of hidden information connected correlated states. Information appears phenomenon of interaction cutting correlations carrying entropy. Each inter action models Kronicker impulse, delta impulse models interaction between the Kronicker impulses. Each impulse step down action cuts maximum of impulse minimal entropy and impulse step up action transits cutting minimal entropy to each step up action of merging delta function. Delta step down action kills delivering entropy producing equivalent minimax information. The merging action initiates quantum microprocess. Multiple cutting entropy is converting to information micro macroprocess. Cutting impulse entropy integrates entropy functional EF along trajectories of multidimensional diffusion process. Information which delivers ending states of each impulse integrates information path functional IPF along process trajectories. Hidden information evaluates Feller kernel whose minimal path transforms Markov transition probability to probability of Brownian diffusion. Each transitive transformation virtually observes origin of hidden information probabilities correlated states. IPF integrates observing Bits along minimal path assembling information Observer. Minimax imposes variation principle on EF and IPF whose extreme equations describe observing micro and macroprocess which describes irreversible thermodynamics. Hidden information curries free information frozen from correlated connections. Free information binds observing micro macro processes in information macrodynamics. Each dynamic three free information composes triplet structures. Three structural triplets assemble information network. Triple networks free information cooperate information Observer.

arXiv:1207.3091•Jul 12, 2012•1 citation
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.
Hidden information emerges under impulse interactions with Markov diffusion process modeling interactive random environment. Impulse yes no action cuts Markov correlations revealing Bit of hidden information connected correlated states. Information appears phenomenon of interaction cutting correlations carrying entropy. Each inter action models Kronicker impulse, delta impulse models interaction between the Kronicker impulses. Each impulse step down action cuts maximum of impulse minimal entropy and impulse step up action transits cutting minimal entropy to each step up action of merging delta function. Delta step down action kills delivering entropy producing equivalent minimax information. The merging action initiates quantum microprocess. Multiple cutting entropy is converting to information micro macroprocess. Cutting impulse entropy integrates entropy functional EF along trajectories of multidimensional diffusion process. Information which delivers ending states of each impulse integrates information path functional IPF along process trajectories. Hidden information evaluates Feller kernel whose minimal path transforms Markov transition probability to probability of Brownian diffusion. Each transitive transformation virtually observes origin of hidden information probabilities correlated states. IPF integrates observing Bits along minimal path assembling information Observer. Minimax imposes variation principle on EF and IPF whose extreme equations describe observing micro and macroprocess which describes irreversible thermodynamics. Hidden information curries free information frozen from correlated connections. Free information binds observing micro macro processes in information macrodynamics. Each dynamic three free information composes triplet structures. Three structural triplets assemble information network. Triple networks free information cooperate information Observer.