Finding dilations of Markov semigroups to cocycle perturbations of noises is a fundamental problem of quantum probablity and quantum dynamics. A noise is hereby a reversible quantum dynamics with a filtration that is independent in some notion of quantum independence. The noise is a dilation of the trivial Markov semigroup; the scope is turning it into a dilation of a nontrivial Markov semigroup by perturbation with a unitary cocycle. In 2009, we proved that a Markov semigroup admits a dilation to a cocycle perturbation of a noise if (and, obviously, only if) it is spatial. A Markov semigroup is spatial if it ”dominates” an ”elementary” CP-semigroup. Spatiality of a Markov semigroup is reflected by its GNS product system and, in fact, in the construction of the dilation the GNS-system and the classification of E0-semigroups by their product systems up to ”stable” cocycle conjugacy plays a crucial role. Spatiality is a concept that occurs on all levels of quantum dynamics: On the level of Markov (or, more generally, CP-) semigroups; on the level of E0-semigroups (that is, unital endomorphism semigroups) no matter if dilating or not dilating a Markov semigroup; and on the level of product systems. Product systems arise (in different ways) from Markov and E0- semigroups; they help to classify dynamics and to construct dilations. In this talk we wish to discuss spatiality on all these levels in general, and how the three levels are intervowen. As an application, we wish to arrive at our result about equivalence of spatiality of a Markov semigroup and existence of a dilation via cocycle perturbation of a noise. We will mention only briefly our result from 2004 that, in a sense, every noise has a filtration that is conditional (or amalgamated, or operator-valued) monotone independent; it is a monotone noise. But there may be filtrations that are independent in other notions of independences, for instance, in conditional (or amalgamated) free independence. Our construction of a Dilation as cocycle perturbation is ”general abstract nonsense”. In particular, it is unclear if the unitary cocycle is adapted to a suitable independent filtration or if it fulfills a quantum stochastic differential equation. These questions will be addressed more specifically in my talk in Bengaluru in two weeks. There I will illustrate how spatial product systems give rise to free product systems. The latter can be used to construct free noises that are good candidates for allowing adapted cocycle perturbations that dilate spatial Markov semigroups. An example is the dilation of a uniformly continuous Markov semigroup on the Pimsner-Speicher full Fock module with the help of my quantum stochastic calculus.

Spatial Quantum Dynamics

SKEIDE, Michael
2012-01-01

Abstract

Finding dilations of Markov semigroups to cocycle perturbations of noises is a fundamental problem of quantum probablity and quantum dynamics. A noise is hereby a reversible quantum dynamics with a filtration that is independent in some notion of quantum independence. The noise is a dilation of the trivial Markov semigroup; the scope is turning it into a dilation of a nontrivial Markov semigroup by perturbation with a unitary cocycle. In 2009, we proved that a Markov semigroup admits a dilation to a cocycle perturbation of a noise if (and, obviously, only if) it is spatial. A Markov semigroup is spatial if it ”dominates” an ”elementary” CP-semigroup. Spatiality of a Markov semigroup is reflected by its GNS product system and, in fact, in the construction of the dilation the GNS-system and the classification of E0-semigroups by their product systems up to ”stable” cocycle conjugacy plays a crucial role. Spatiality is a concept that occurs on all levels of quantum dynamics: On the level of Markov (or, more generally, CP-) semigroups; on the level of E0-semigroups (that is, unital endomorphism semigroups) no matter if dilating or not dilating a Markov semigroup; and on the level of product systems. Product systems arise (in different ways) from Markov and E0- semigroups; they help to classify dynamics and to construct dilations. In this talk we wish to discuss spatiality on all these levels in general, and how the three levels are intervowen. As an application, we wish to arrive at our result about equivalence of spatiality of a Markov semigroup and existence of a dilation via cocycle perturbation of a noise. We will mention only briefly our result from 2004 that, in a sense, every noise has a filtration that is conditional (or amalgamated, or operator-valued) monotone independent; it is a monotone noise. But there may be filtrations that are independent in other notions of independences, for instance, in conditional (or amalgamated) free independence. Our construction of a Dilation as cocycle perturbation is ”general abstract nonsense”. In particular, it is unclear if the unitary cocycle is adapted to a suitable independent filtration or if it fulfills a quantum stochastic differential equation. These questions will be addressed more specifically in my talk in Bengaluru in two weeks. There I will illustrate how spatial product systems give rise to free product systems. The latter can be used to construct free noises that are good candidates for allowing adapted cocycle perturbations that dilate spatial Markov semigroups. An example is the dilation of a uniformly continuous Markov semigroup on the Pimsner-Speicher full Fock module with the help of my quantum stochastic calculus.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11695/18111
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