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Journal of Mathematical Physics

Quantum stochastic processes for maps on Hilbert C*-modules

Jaeseong Heo, Un Cig Ji

Published May 1, 2011• 10 citations
Publisher

Abstract

We discuss pairs (ϕ, Φ) of maps, where ϕ is a map between C*-algebras and Φ is a ϕ-module map between Hilbert C*-modules, which are generalization of representations of Hilbert C*-modules. A covariant version of Stinespring's theorem for such a pair (ϕ, Φ) is established, and quantum stochastic processes constructed from pairs ({ϕt}, {Φt}) of families of such maps are studied. We prove that the quantum stochastic process J = {Jt} constructed from a ϕ-quantum dynamical semigroup Φ = {Φt} is a j-map for the quantum stochastic process j = {jt} constructed from the given quantum dynamical semigroup ϕ = {ϕt}, and that J is covariant if the ϕ-quantum dynamical semigroup Φ is covariant.

DOI
10.1063/1.3582778

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We discuss pairs (ϕ, Φ) of maps, where ϕ is a map between C*-algebras and Φ is a ϕ-module map between Hilbert C*-modules, which are generalization of representations of Hilbert C*-modules. A covariant version of Stinespring's theorem for such a pair (ϕ, Φ) is established, and quantum stochastic processes constructed from pairs ({ϕt}, {Φt}) of families of such maps are studied. We prove that the quantum stochastic process J = {Jt} constructed from a ϕ-quantum dynamical semigroup Φ = {Φt} is a j-map for the quantum stochastic process j = {jt} constructed from the given quantum dynamical semigroup ϕ = {ϕt}, and that J is covariant if the ϕ-quantum dynamical semigroup Φ is covariant.
Based on the Hilbert C∗C^*C∗ -module structure we study the reconstruction theorem for stationary monotone quantum Markov processes from quantum dynamical semigroups. We prove that the quantum stochastic monotone process constructed from a covariant quantum dynamical semigroup is again covariant in the strong sense.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
In this work, we investigate the existence of a factorization for a unital completely positive map, between non-commutative probability space which do not change the expectation values of the events. These maps are called in literature stochastic maps. Using the Stinespring representations of completely positive map and assuming the existence of anti-unitary operator on Hilbert space related to these representations which satisfying some modular relations, we prove that stochastic maps with adjoint, admits a factorization.
Statistically interpretable axioms are formulated that define a quantum stochastic process (QSP) as a causally ordered operator field in an arbitrary space-time region T of an open quantum system under a sequential observation at a discrete space-time localization. It is shown that to every QSP described in the weak sense by a self-consistent system of causally ordered correlation kernels there corresponds a unique, up to unitary equivalence, minimal QSP in the strong sense. It is shown that the proposed QSP construction, which reduces in the case of the linearly ordered discrete T=Z to the construction of the inductive limit of Lindblad's canonical representations, corresponds to Kolmogorov's classical reconstruction if the order on T is ignored and leads to Lewis construction if one uses the system of all (not only causal) correlation kernels, regarding this system as lexicographically preordered on T. The approach presented encompasses both nonrelativistic and relativistic irreversible dynamics of open quantum systems and fields satisfying the conditions of local commutativity and semigroup covariance. Also given are necessary and sufficient conditions of dynamicity (or conditional Markovianity) and regularity, these leading to the properties of complete mixing (relaxation) and ergodicity of the QSP.