Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Free Product Systems

Michael Skeide

Published Jan 1, 2011• 0 citations
OpenAlex

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 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.

Related papers

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

Dilation of Markovian cocycles on a von Neumann algebra

Debashish Goswami, J. Martin Lindsay, Kalyan B. Sinha +1 more

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.

Pacific Journal of Mathematics•Oct 1, 2003•18 citations

Operator Markovian Cocycles via Associated Semigroups

J. Martin Lindsay, Stephen J. Wills

Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary

•Dec 16, 2005•1 citation

Mini-Workshop: Product Systems and Independence in Quantum Dynamics

Mathematisches Forschungsinstitut Oberwolfach

Abstract. Quantum dynamics, both reversible (i.e., closed quantum sys-tems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in par-ticular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally in-dependent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or mod-ule determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.

•Jan 1, 2009•0 citations

Mini-Workshop: Product Systems and Independence in Quantum Dynamics

B. V. Rajarama Bhat, Uwe Franz, Michael Skeide

Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.

Oberwolfach Reports•Dec 22, 2009•6 citations

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.

Cambridge University Press eBooks•Feb 9, 2015•0 citations
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.
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.)
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.
Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary
Abstract. Quantum dynamics, both reversible (i.e., closed quantum sys-tems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in par-ticular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally in-dependent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or mod-ule determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.
Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a filtration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this field.
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.