Dilation of a class of quantum dynamical semigroups with unbounded generators on UHF algebras

Abstract

Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in [Rev. Math. Phys. 5 (3) (1993) 587-600]. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.

A Homomorphism Theorem and a Trotter Product Formula for Quantum Stochastic Flows with Unbounded Coefficients

“…In [5] the authors considered the problem of constructing quantum stochastic dilations of quantum dynamical semigroups arising as above. However they could construct quantum stochastic dilations for only a special class of semigroups, namely those for which the associated completely positive map ψ is of the form: ψ(x) = r * xr for r := g∈ j∈Z d Z N c g W g with W g := j∈Z d (U a V b ) α j , where g := j∈Z d α j such that g |c g ||g| 2 < ∞, a, b ∈ Z N being fixed.…”

“…Proof. The existence of the dilation (j (m) t ) t≥0 is the main result in [5]. Since the flow (j (m) t ) t≥0 is a * -homomorphic flow for each m = 1, 2, ..., p, by Lemma 2.9 the flow satisfies assumption A(1).…”

“…, where {e j } j is an orthonormal basis for the associated noise space k 0 (see [5]). By generates a C 0 -semigroup on L 1 (tr) and moreover, A 0 is a core for this map as well.…”

An algebraic construction of quantum flows with unbounded generators

“…In Section 6 we use Theorem 3.12 to obtain flows on some universal C * algebras, namely the noncommutative torus and the universal rotation algebra [2]; the former is a particularly important example in non-commutative geometry. Quantum flows on these algebras have previously been considered by Goswami, Sahu and Sinha [14] and by Hudson and Robinson [17], respectively.…”

“…This object, a model for systems of interacting quantum particles, was introduced by Rebolledo [27] as a non-commutative generalisation of the classical exclusion process [18] and has generated much interest: see [13] and [12]. The multiplicity space k is required to be infinite dimensional for this process, as in previous work on processes arising from quantum interacting particle systems, e.g., [14].…”