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arXiv:0809.3538 · Symmetry Integrability and Geometry Methods and Applications

Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

Universit&, Michael Skeide, #224, degli Studi del Molise, Italy

Published Oct 3, 2022• 2 citations
arXivHTMLPDF

Abstract

We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on B(G) (G a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup).The proof is by general abstract nonsense (taken from Arveson's classification of E 0 -semigroups on B(H) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy.All other results that make similar statements (especially, [Mem.Amer.Math.Soc.240 (2016), vi+126 pages, arXiv:0901.1798]) for more general C * -algebras) have been proved later by suitable adaptations of the methods exposed here.(They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)

DOI
10.3842/sigma.2022.071

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We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on B(G) (G a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup).The proof is by general abstract nonsense (taken from Arveson's classification of E 0 -semigroups on B(H) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy.All other results that make similar statements (especially, [Mem.Amer.Math.Soc.240 (2016), vi+126 pages, arXiv:0901.1798]) for more general C * -algebras) have been proved later by suitable adaptations of the methods exposed here.(They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)
Hudson-Parthasarathy (H-P) type quantum stochastic dilation of a class of C 0 semigroups of completely positive maps ( quantum dynamical or Markov semigroups) on a von Neumann or C * algebra, with unbounded generators, is constructed under some assumptions on the semigroup and its generator.The assumption of symmetry with respect to a semifinite trace allows the use of Hilbert space techniques,while that of covariance with respect to an action of a Lie group on the algebra gives a better control on the domain of the generator.A dilation of the dynamical semigroup is obtained, under some further assumptions on the domain of the generator, with the help of a conjugation by a unitary quantum stochastic process satisfying Hudson-Parthasarathy equation in Fock space.
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 that is turned into a dilation of a nontrivial Moarkov semigroup by perturbation with a unitary cocycle. In 2004, we pointed out 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 filtration that are independent on other notions of independences, for instance, in conditional (or amalgamated) free independence. 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 as 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. But this construction is "gereral abstract nonsense". In particular, it is unclear if the unitrary cocycle is adapted to a suitable independent filtration or if it fulfills a quantum stochastic differential equation. After recalling some details about these facts, we introduce free product systems an illustrate how to contruct a conditional (or amalgamated) free noise from every free productr system. We show how every spatial (tensor!) product system generates a free product system. For instance, as we pointed out in 2001, the time ordered product systems (that is, the spatial type I systems) generate the free product system of full Fock modules with the same index. For uniformly continuous continuous Markov semigroups, we constructed suitable unitary cocycles with our Quantum stochastic calculus on the full Fock module from 2000. Here, we ask if it is possible to do the same, purely algebraically without any caclulus, starting from the GNS-systsem of an arbitrary spatial Markov semigroup.
A covariant version of the dilation theory for quantum dynamical semigrops is established, including both Bhat-Parthasarathy type and Evans-Hudson type dilation. It is shown in particular that every uniformly continuous quantum dynamical semigroup on a separable C*-algebra or a von Neumann algebra, which is covariant under a suitable group-action, admits a covariant Evans-Hudson dilation in some appropriate Fock space. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract. We consider normal Markovian cocycles on a von Neumann al-gebra which are adapted to a Fock filtration. Every such cocycle k which is Markov-regular and consists of completely positive contractions is realised as a conditioned ∗-homomorphic cocycle. This amounts to a stochastic generalisa-tion of a recent dilation result for norm-continuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coef-ficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of cocycles. Number/exchange-free dilatability is seen to be related to locality in the case where the cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markov-regular contraction cocycles on a Hilbert space are also described. The paper ends with a discussion of connections with measure-valued diffusion.
We study the Classical Probability analogue of the unitary dilations of a quantum dynamical semigroup defined in Quantum Probability via quantum stochastic differential equations. Given a homogeneous Markov chain in continuous time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the unitary dilations of a quantum dynamical semigroup in Quantum Probability.
A well-known theorem of W. Arveson states that a completely positive (CP) map dominated (difference is CP) by a given CP map is described through a positive contraction in the commutant of the homomorphism of Stinespring representation. This contraction plays the role of Radon-Nikodym derivative of measure theory. We look at implications of this to one parameter semigroups of unital completely positive maps, known as quantum Markov semigroups. We focus our attention to quantum Markov semigroups of B(H), the algebra of all bounded operators on a Hilbert space. These semigroups can be dilated to semigroups of unital endomorphisms (E₀-semigroups) of B(K) for some other Hilbert space K. Here we see that dominated CP semigroups dilate to dominated semigroups of the dilation. Moreover at the level of Eo-semigroups the dominated CP semigroups are described through positive contractive local cocycles, which are really families of contractions of 'Radon-Nikodym derivatives' mentioned above. We analyze the structure for quantum semigroups with bounded generators.