Restrictions of CP-semigroups to maximal commutative subalgebras

Abstract

Abstract. We give a necessary and sufficient criterion for a normal CP-map on a von Neumann algebra to admit a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on B(G).

On the generators of quantum dynamical semigroups with invariant subalgebras

“…According to Theorem 4, we can reduce the above question to characterizing CP-maps with an invariant maximal abelian vN-subalgebra. The latter question was previously investigated in [11,23]. More precisely, [23,Theorem 1] gave an abstract characterization of such GKLS-generators in terms of a commutation relation and a sufficient condition on the Kraus operators of the CP part of the GKLS-generator.…”

“…Ref. [11] extended these deliberations and, in particular, gave a necessary and sufficient condition for a normal CP-map to leave a maximal abelian vN-subalgebra invariant: The "if"-direction in Theorem 10 can be seen using the von Neumann bicommutant theorem. We can recover the "only if"-direction, albeit only for atomic C, as a consequence of Theorem 5 as follows: If C is a maximal abelian and atomic vN-subalgebra of L(H), then its decomposition as in Definition 3 becomes particularly simple, with dim(H A i ) = 1 = dim(H B i ) for all i ∈ I.…”

“…Among the results for which the answer to the question posed above is useful are: The Koashi-Imoto theorem [16], an important result in the theory of quantum communication, giving the general form of a quantum channel leaving a certain set of density matrices invariant; the form of the GKLS-generator imposed by the invariance of the decoherence-free subalgebra [1,7,10,24]; general questions about decoherence, where the form of the GKLS-generator imposed by the invariance of a maximally abelian subalgebra is important [5,11,[21][22][23]; the study of Markovian subsystems [26] and the study of the aging process of quantum devices via dynamical semigroups of superchannels [13].…”

“…[11, Corollary 3.4] Let Φ be a normal CP-map on L(H) with Kraus decomposition Φ(X) = n∈N φ † n Xφ n . Let C be a maximal abelian vN-subalgebra of L(H).…”

Quantum extensions of dynamical systems and of Markov semigroups

“…The question of the existence of such a stable algebra was first arised and motivated by Rebolledo in [2] and then studied by several authors ( [3], [4]). We shall be concerned with the inverse problem: given a classical Markov semigroup, is it the restriction of a quantum Markov semigroup to a maximal stable commutative algebra?…”