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arXiv:math/0505384

A resolution of quantum dynamical semigroups

Anilesh Mohari

Published May 18, 2005• 3 citations
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Abstract

We consider a class of quantum dissipative systems governed by a one parameter completely positive maps on a von-Neumann algebra. We introduce a notion of recurrent and metastable projections for the dynamics and prove that the unit operator can be decomposed into orthogonal projections where each projections are recurrent or metastable for the dynamics.

DOI
10.48550/arxiv.math/0505384

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We consider a class of quantum dissipative systems governed by a one parameter completely positive maps on a von-Neumann algebra. We introduce a notion of recurrent and metastable projections for the dynamics and prove that the unit operator can be decomposed into orthogonal projections where each projections are recurrent or metastable for the dynamics.
Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following C*-algebraic setting: A given Hermitian dissipative mapping δ is densely defined in a unital C*-algebra A. The identity element in A is also in the domain of δ. Completely dissipative maps δ are defined by the requirement that the induced maps, (aij)→(δ(aij)), are dissipative on the n-by-n complex matrices over A for all n. We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of A is injective, we show the existence of an extension of δ which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If δ is a given well-behaved *-derivation, then we show that each of the maps ±δ is completely dissipative.
We consider hermitian dissipative mappings δ which are densely defined in unital C ∗-algebras A. The identity element in A is also in the domain of δ. Completely dissipative maps δ are defined by the requirement that the induced maps, (aij) → (δ(aij)), are dissipative on the n by n complex matrices over A for all n. We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of A is injective, we show the existence of an extension of δ which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If δ is a given well-behaved ∗-derivation, then we show that each of the maps ±δ is completely dissipative. 1.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).
We generalize the result of Gorini, Kossakowski, and Sudarshan [J. Math. Phys. 17:821, 1976] that every generator of a quantum-dynamical semigroup decomposes uniquely into a closed and a dissipative part, assuming the trace of both vanishes. More precisely, we show that given any generator [Formula: see text] of a completely positive dynamical semigroup and any matrix [Formula: see text] there exists a unique matrix [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] has trace zero, and (iii) [Formula: see text] is a real number. The key to proving this is the relation between the trace of a completely positive map, the trace of its Kraus operators, and expectation values of its Choi matrix. Moreover, we show that the above decomposition is orthogonal with respect to some [Formula: see text]-weighted inner product.
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 ≤ ∈.