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Comparison of completely positive maps on a C* algebra and a Lebesgue decomposition theorem

Kalyanapuram Rangachari Parthasarathy

Lecture notes in statisticsJan 1, 199610 citations

A continuity theorem for Stinespring's dilation

Dennis Kretschmann, Dirk Schlingemann, Reinhard F. Werner

arXiv:0710.2495Journal of Functional AnalysisSep 11, 200861 citations

Quantum Stochastic Calculus

Kalyanapuram Rangachari Parthasarathy

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.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.

Birkhäuser Basel eBooksJan 1, 19956 citations

A review on semigroups of completely positive maps

David E. Evans

Springer eBooksApr 6, 20089 citations

Chapter Three. Completely Positive Maps

Princeton University Press eBooksDec 31, 20090 citations

A covariant Stinespring theorem

Dominic Verdon

We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.We prove a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T≔Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X:T→M1, Y:T→M2, we show that covariant completely positive maps between the corresponding G-C*-algebras can be “dilated” to isometries τ: X → Y ⊗ E, where E:M2→M1 is some “environment” 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimizing the quantum dimension of the environment is unique up to a unitary. When G is a compact group, this recovers previous covariant Stinespring-type theorems.

arXiv:2108.09872Journal of Mathematical PhysicsSep 1, 20228 citations

A continuous time version of Stinespring’s theorem on completely positive maps

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