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Covariant Markov Dilations of Quantum ^55^

A. Frigerio

Published Jan 1, 1985• 0 citations
OpenAlex

Abstract

We develop a general theory of unitary dilations of quantum dynamical semigroups, satisfying a Markov property; and we construct explicit examples by means of the noncommutative stochastic calculus of Hudson and Parthasarathy.

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Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations

Universit&, Michael Skeide, #224 +1 more

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.)

arXiv:0809.3538•Symmetry Integrability and Geometry Methods and Applications•Oct 3, 2022•2 citations
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 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.
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.
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.)