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Classical and Quantum Semigroups

Franco Fagnola

Published Apr 4, 2018• 0 citations
Publisher

Abstract

Several quantum Markov semigroups admit an invariant abelian subalgebra which can be looked at as an algebra of bounded functions on some measurable space. The restriction to this subalgebra is a semigroup of positive and identity preserving operators i.e. a classical Markov semigroup (see Examples 3.3, 3.4).
DOI
10.22199/s07160917.1999.0003.00005

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Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad \cite{lindblad}, Stinespring \cite{stinespring}, and Kraus \cite{kraus} imply that the generator of the semigroup has the form L(A)=∑n=1∞Vn∗AVn+GA+AG∗L(A)=\sum_{n=1}^{\infty}V_n^*AV_n +GA+AG^*L(A)=∑n=1∞​Vn∗​AVn​+GA+AG∗ where VnV_nVn​ and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.
Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.
Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to the works of Lindblad [Commun. Math. Phys. 48, 119-130 (1976)], Stinespring [Proc. Am. Math. Soc. 6, 211-216 (1955)], and Kraus [Ann. Phys. 64, 311-335 (1970)] imply that the generator of the semigroup has the form L(A)=∑n=1∞Vn∗AVn+GA+AG∗, where Vn and G are elements of the underlying operator algebra. In the present paper, we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus’ result. Finally, we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.
Abstract We demonstrate a method for finding the decoherence-free subalgebra N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra B(Γ(Cd)){\mathcal {B}}(\Gamma (\mathbb {C}^d))B(Γ(Cd)) B ( Γ ( C d ) ) of all bounded operator on the Fock space Γ(Cd)\Gamma (\mathbb {C}^d)Γ(Cd) Γ ( C d ) on Cd\mathbb {C}^dCd C d . We show that N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) is a type I von Neumann algebra L∞(Rdc;C)⊗‾B(Γ(Cdf))L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))L∞(Rdc​;C)⊗​B(Γ(Cdf​)) L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers dc,df≤dd_c,d_f\le ddc​,df​≤d d c , d f ≤ d . This result is illustrated by some applications and examples.