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28
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Chaotic states and Stochastic Integration in Quantum Systems
 Russian Math. Survey
, 1992
"... Abstract. Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the simple monoidvalued fields on an atomless ‘spacetime’ ..."
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Cited by 14 (12 self)
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Abstract. Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the simple monoidvalued fields on an atomless ‘spacetime’ set. A canonical decomposition of the logarithmic conditionally posivedefinite generating functional is constructed in a pseudoEuclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudonorm state in this space. It is shown that the exponential representation in the corresponding pseudoFock space yields the infinitelydivisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitlydivisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Itô algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly
Quantum Probability applied to the Damped Harmonic Oscillator
"... Contents 1. The Framework of Quantum Probability 1.1. Making probability noncommutative 1.2. Events and random variables 1.3. Interpretation of quantum probability 1.4. The quantum coin toss: `spin' 1.5. Positive denite kernels 2. Some Quantum Mechanics 2.1. Position and momentum 2.2. Energy and ..."
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Cited by 9 (0 self)
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Contents 1. The Framework of Quantum Probability 1.1. Making probability noncommutative 1.2. Events and random variables 1.3. Interpretation of quantum probability 1.4. The quantum coin toss: `spin' 1.5. Positive denite kernels 2. Some Quantum Mechanics 2.1. Position and momentum 2.2. Energy and time evolution 2.3. The harmonic oscillator 2.4. The problem of damping 3. Conditional Expectations and Operations 3.1. Conditional expectations in nite dimension 3.2. Operations in nite dimension 3.3. Operations on quantum probability spaces 3.4. Quantum stochastic processes 3.5. Conditional expectations and transition operators 3.6. Markov processes 4. Second Quantisation 4.1. The functor 4.2. Fields 4.3. Quanta 5. Unitary dilations of spiralling motion 6. The Damped Harmonic Oscillator 6.1. Stochastic behaviour of the oscillator 6.2. The driving eld 6.3. Excitations of the oscillator 6.4. Emitted quanta
Continual measurements in quantum mechanics and quantum stochastic calculus
 in Open Quantum Systems III: Recent Developments
, 2006
"... 1.1 Three approaches to continual measurements................... 3 1.2 Quantum stochastic calculus and quantum optics................ 3 1.3 Some notations: operator spaces............................... 4 2 Unitary evolution and states.............................. 5 ..."
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Cited by 8 (0 self)
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1.1 Three approaches to continual measurements................... 3 1.2 Quantum stochastic calculus and quantum optics................ 3 1.3 Some notations: operator spaces............................... 4 2 Unitary evolution and states.............................. 5
On the Algebraic Foundations of NonCommutative Probability Theory
"... As a first part of a rigorous mathematical theory of noncommutative probability we present, starting from a set of canonical axioms, a complete classification of the notions of noncommutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schurmann ..."
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Cited by 5 (0 self)
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As a first part of a rigorous mathematical theory of noncommutative probability we present, starting from a set of canonical axioms, a complete classification of the notions of noncommutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schurmann [21; 22] and is based on a fundamental paper by R. Speicher [26]. Our concept is used to initiate a theory of noncommutative L'evy processes which are defined on dual groups in the sense of D. Voiculescu [29]. The paper generalises L'evy processes on Hopf algebras [20] to noncommutative independences other than the `tensor' independence of R. L. Hudson [9, 12].
A Scattering Theory for Markov Chains
, 2000
"... This paper is organised as follows. Sections 1 and 2 give the necessary background. Sections 3 and 4 develop criteria for the existence of scattering operators. They are applied to various situations in Sections 5, 6 and 7. x1. Probability spaces and stochastic processes. ..."
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Cited by 5 (0 self)
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This paper is organised as follows. Sections 1 and 2 give the necessary background. Sections 3 and 4 develop criteria for the existence of scattering operators. They are applied to various situations in Sections 5, 6 and 7. x1. Probability spaces and stochastic processes.
2000]: “Classical and quantum probability
 Journ. Math. Phys
"... We follow the development of probability theory from the beginning of the last century, emphasising that quantum theory is really a generalisation of this theory. The great achievements of probability theory, such as the theory of processes, generalised random fields, estimation theory and informati ..."
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Cited by 5 (0 self)
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We follow the development of probability theory from the beginning of the last century, emphasising that quantum theory is really a generalisation of this theory. The great achievements of probability theory, such as the theory of processes, generalised random fields, estimation theory and information geometry, are reviewed. Their quantum versions are then described.
An algebraic approach to the KolmogorovSinai entropy
, 1995
"... We revisit the notion of KolmogorovSinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for defining the entropy for general noncommutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynam ..."
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Cited by 5 (1 self)
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We revisit the notion of KolmogorovSinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for defining the entropy for general noncommutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, selfcontained proof of the entropy formula for general automorphisms of ndimensional tori. Mathematics Subject Classification (1991): 28D20, 47A35, 46L55, 82B10 1 Email: fizra@halina.univ.gda.pl 2 Email: johan.andries@fys.kuleuven.ac.be 3 Onderzoeksleider NFWO Belgium, Email: mark.fannes@fys.kuleuven.ac.be 4 Onderzoeker IIKW, Email: pim.tuyls@fys.kuleuven.ac.be 1. Introduction The aim of this paper is to investigate the compatibility of the construction of the dynamical entropy for general noncommutative dynamical systems, as proposed in [AF1], with the wellestablished notion of entropy for classical d...
Interactions in noncommutative dynamics
 Comm. Math. Phys
, 2000
"... Abstract. For a fixed C ∗algebra A, we consider all noncommutative dynamical systems that can be generated by A. More precisely, an Adynamical system is a triple (i, B, α) where α is a ∗endomorphism of a C ∗algebra B, and i: A ⊆ B is the inclusion of A as a C ∗subalgebra with the property that ..."
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Cited by 4 (4 self)
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Abstract. For a fixed C ∗algebra A, we consider all noncommutative dynamical systems that can be generated by A. More precisely, an Adynamical system is a triple (i, B, α) where α is a ∗endomorphism of a C ∗algebra B, and i: A ⊆ B is the inclusion of A as a C ∗subalgebra with the property that B is generated by A ∪α(A) ∪α 2 (A) ∪ · · ·. There is a natural hierarchy in the class of Adynamical systems, and there is a universal one that dominates all others, denoted (i, PA,α). We establish certain properties of (i, PA,α) and give applications to some concrete issues of noncommutative dynamics. For example, we show that every contractive completely positive linear map ϕ: A → A gives rise to to a unique Adynamical system (i, B, α) that is “minimal ” with respect to ϕ, and we show that its C ∗algebra B can be embedded in the multiplier algebra of A ⊗ K. 1. Generators The flow of time in quantum theory is represented by a oneparameter group of ∗automorphisms {αt: t ∈ R} of a C ∗algebra B. There is often a C ∗subalgebra A ⊆ B that can be singled out from physical considerations which, together with its time translates, generates B. For example, in nonrelativistic quantum mechanics the flow of time is represented by a oneparameter group of automorphisms of B(H), and the set of all bounded continuous functions of the configuration observables at time 0 is a commutative C ∗algebra A. The set of all time translates αt(A) of A generates an irreducible C∗subalgebra B of B(H). In particular, for different times t1 ̸ = t2, the C∗algebras αt1 (A) and αt2 (A) do not commute with each other. Indeed, no nontrivial relations appear to exist between αt1 (A) and αt2 (A) when t1 ̸ = t2. In this paper we look closely at this phenomenon, in a simpler but analogous setting. Let A be a C∗algebra, fixed throught. Definition 1.1. An Adynamical system is a triple (i,B,α) consisting of a ∗endomorphism α acting on a C ∗algebra B and an injective ∗homomorphism i: A → B, such that B is generated by i(A) ∪ α(i(A)) ∪ α 2 (i(A)) ∪ · · ·. We lighten notation by identifying A with its image i(A) in B, thereby replacing i with the inclusion map i: A ⊆ B. Thus, an Adynamical system is a dynamical system (B,α) that contains A as a C ∗subalgebra in