A stochastic Stinespring Theorem

Alfredo González Lezcano, Alejandro Cabo Montes de

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.

Jaeseong Heo, Un Cig Ji

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.

A. O. Bolivar

Dennis Kretschmann, Dirk-M. Schlingemann, Reinhard F. Werner

Stinespring's dilation theorem is the basic structure theorem for quantum channels: it states that any quantum channel arises from a unitary evolution on a larger system. Here we prove a continuity theorem for Stinespring's dilation: if two quantum channels are close in cb-norm, then it is always possible to find unitary implementations which are close in operator norm, with dimension-independent bounds. This result generalizes Uhlmann's theorem from states to channels and allows to derive a formulation of the information-disturbance tradeoff in terms of quantum channels, as well as a continuity estimate for the no-broadcasting theorem. We briefly discuss further implications for quantum cryptography, thermalization processes, and the black hole information loss puzzle.