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Cambridge University Press eBooks

Quantum dynamical semigroups

Kalyan B. Sinha, Debashish Goswami

Published Jan 25, 2007• 0 citations
Publisher

Abstract

Let us now restrict ourselves to the case when the general locally convex space X is replaced by a C * or a von Neumann algebra A , and study the implications of the complete positivity of a semigroup T t acting on it. Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C * -algebra A is a contractive semigroup T t of class C 0 such that each T t is a completely positive map from A to itself. T t is said to be conservative if T t (1) = 1 for all t ≥ 0. Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans For a uniformly continuous semigroup on a von Neumann algebra A ⊆ B ( h ), we have the following result. Lemma 3.1.1 Let T t = e tL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then T t is normal for each t if and only if L is ultra-strongly (and hence ultra-weakly) continuous on any norm-bounded subset of A . Proof : Let us first note that L is norm-bounded. If L is ultra-strongly continuous on bounded sets, then clearly e tL is ultra-strongly continuous on bounded sets for each t , and hence normal. For the converse, first note that for any t ≥ 0 and x ∈ A , we have Hence it is not difficult to see that Now suppose that x α is a net of elements in A such that x α strongly converges to x ∈ A and there exists positive constant M such that ∥ x α ∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t 0 small enough so that ∥ L ∥ 2 M ∥ u ∥ t 0 ≤ ∈.

DOI
10.1017/cbo9780511618529.005

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Let us now restrict ourselves to the case when the general locally convex space X is replaced by a C * or a von Neumann algebra A , and study the implications of the complete positivity of a semigroup T t acting on it. Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C * -algebra A is a contractive semigroup T t of class C 0 such that each T t is a completely positive map from A to itself. T t is said to be conservative if T t (1) = 1 for all t ≥ 0. Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans For a uniformly continuous semigroup on a von Neumann algebra A ⊆ B ( h ), we have the following result. Lemma 3.1.1 Let T t = e tL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then T t is normal for each t if and only if L is ultra-strongly (and hence ultra-weakly) continuous on any norm-bounded subset of A . Proof : Let us first note that L is norm-bounded. If L is ultra-strongly continuous on bounded sets, then clearly e tL is ultra-strongly continuous on bounded sets for each t , and hence normal. For the converse, first note that for any t ≥ 0 and x ∈ A , we have Hence it is not difficult to see that Now suppose that x α is a net of elements in A such that x α strongly converges to x ∈ A and there exists positive constant M such that ∥ x α ∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t 0 small enough so that ∥ L ∥ 2 M ∥ u ∥ t 0 ≤ ∈.
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In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.
In Chapter 6 we built a theory of quantum stochastic dilation ‘naturally’ associated with an arbitrary Q.D.S. on a von Neumann or C * -algebra with bounded generator. There the computations involved C * or von Neumann Hilbert modules, using the results of [24], map-valued quantum stochastic processes on modules and quantum stochastic integration with respect to them, developed in Chapter 5. It is now natural to consider the case of a Q.D.S. with unbounded generator and ask the same questions about the possibility of dilation. As one would expect, the problem is too intractable in this generality and we need to impose some further structures on it. In this chapter we shall consider a few classes of such Q.D.S. and try to construct H–P and E–H dilation for them. At first, we shall work under the framework of a Lie group action on the underlying algebra, and consider covariant Q.D.S. For H–P dilation, symmetry with respect to a trace is also assumed, whereas a general theory for E–H dilation has been built under the assumption of covariance under the action of a compact group, but without it being symmetric. Then, in the last section, we deal with a class of Q.D.S. on the U.H.F. algebra, described in Chapter 3. In this case, E–H dilation is constructed by a direct iteration using some natural estimates. However, what is common to the methods used in constructing dilation for the different kinds of Q.D.S. mentioned above is the use of a natural locally convex topology, in which the generator (unbounded in the norm topology) is continuous.