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Positive linear maps of matrix algebras

W. A. Majewski (2012)

Banach Center Publications

A characterization of the structure of positive maps is presented. This sheds some more light on the old open problem studied both in Quantum Information and Operator Algebras. Our arguments are based on the concept of exposed points, links between tensor products and mapping spaces and convex analysis.

Quantum Itô B*-algebras, their classification and decomposition

V. Belavkin (1998)

Banach Center Publications

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Itô B*-algebra, generalizing the C*-algebra, is defined to include the Banach infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Itô algebra is canonically decomposed...

Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space

Alexander Chebotarev, Dmitry Victorov (1998)

Banach Center Publications

By using F. A. Berezin's canonical transformation method [5], we derive a nonadapted quantum stochastic differential equation (QSDE) as an equation for the strong limit of the family of unitary groups satisfying the Schrödinger equation with singularly degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type...

Quantum symmetries in noncommutative C*-systems

Marcin Marciniak (1998)

Banach Center Publications

We introduce the notion of a completely quantum C*-system (A,G,α), i.e. a C*-algebra A with an action α of a compact quantum group G. Spectral properties of completely quantum systems are investigated. In particular, it is shown that G-finite elements form the dense *-subalgebra of A. Furthermore, properties of ergodic systems are studied. We prove that there exists a unique α-invariant state ω on A. Its properties are described by a family of modular operators σ z z ∈ ℂ acting on . It turns out that ω...

Reduced and extended weak coupling limit

Jan Dereziński, Wojciech De Roeck (2007)

Banach Center Publications

The main aim of our lectures is to give a pedagogical introduction to various mathematical formalisms used to describe open quantum systems: completely positive semigroups, dilations of semigroups, quantum Langevin dynamics and the so-called Pauli-Fierz Hamiltonians. We explain two kinds of the weak coupling limit. Both of them show that Hamiltonian dynamics of a small quantum system interacting with a large resevoir can be approximated by simpler dynamics. The better known reduced weak coupling...

Supporting sequences of pure states on JB algebras

Jan Hamhalter (1999)

Studia Mathematica

We show that any sequence ( φ n ) of mutually orthogonal pure states on a JB algebra A such that ( φ n ) forms an almost discrete sequence in the relative topology induced by the primitive ideal space of A admits a sequence ( a n ) consisting of positive, norm one, elements of A with pairwise orthogonal supports which is supporting for ( φ n ) in the sense of φ n ( a n ) = 1 for all n. Moreover, if A is separable then ( a n ) can be taken such that ( φ n ) is uniquely determined by the biorthogonality condition φ n ( a n ) = 1 . Consequences of this result improving...

Currently displaying 41 – 60 of 66