Adaptive Dynamics and Its Applications to Chaos and NPC Problem Quantum Algorithm for SAT Problem and Quantum Mutual Entropy New Quantum Algorithm for Studying NP-Complete Problems Entanglement, Quantum Entropy and Mutual Information Fundamentals of Quantum Mutual Entropy and Capacity Compound Channels, Transition Expectations, and Liftings Complexities and Their Applications to Characterization of Chaos Complexity, Fractal Dimension for Quantum States Information Dynamics and Its Applications to Optical Communication Processes Information Theoretical Treatments of Genes Some Aspects of Quantum Information Theory and Their Applications to Irreversible Processes On Compound State and Mutual Information in Quantum Information Theory Note on Quantum Probability Quantum Ergodic Channels in Operator Algebras On Open System Dynamics -- An Operator Algebraic Study Dynamical Process in Linear Response Theory Stability of Weiss Ising Model and other papers.Adaptive Dynamics and Its Applications to Chaos and NPC Problem Quantum Algorithm for SAT Problem and Quantum Mutual Entropy New Quantum Algorithm for Studying NP-Complete Problems Entanglement, Quantum Entropy and Mutual Information Fundamentals of Quantum Mutual Entropy and Capacity Compound Channels, Transition Expectations, and Liftings Complexities and Their Applications to Characterization of Chaos Complexity, Fractal Dimension for Quantum States Information Dynamics and Its Applications to Optical Communication Processes Information Theoretical Treatments of Genes Some Aspects of Quantum Information Theory and Their Applications to Irreversible Processes On Compound State and Mutual Information in Quantum Information Theory Note on Quantum Probability Quantum Ergodic Channels in Operator Algebras On Open System Dynamics -- An Operator Algebraic Study Dynamical Process in Linear Response Theory Stability of Weiss Ising Model and other papers.

WORLD SCIENTIFIC eBooksFeb 1, 20080 citations

Quantum stochatic integrals and Doob-Meyer decomposition

Andrzej Łuczak

We show that for a quantum $L^p$ -martingale $(X(t))$ , $p>2$ , there exists a Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$ . A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an $L^p$ -martingale, $p>2$ , is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained.We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an $L^p$-martingale, $p>2$, is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained.

arXiv:math/0602216Feb 10, 20061 citation

Classical and quantum entropy

Benjamin W. Schumacher, Michael D. Westmoreland

Cambridge University Press eBooksMar 25, 20100 citations

Quantum Entropy and Its Use

Masanori Ohya, Dėnes Petz

Springer eBooksJan 1, 1993764 citations

We show that for a quantum

L^p

-martingale

(X(t))

p>2

, there exists a Doob-Meyer decomposition of the submartingale

(|X(t)|^2)

. A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an

L^p

-martingale,

p>2

, is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained.