On Period, Cycles and Fixed Points of a Quantum Channel

Abstract

We consider a quantum channel acting on an infinite dimensional von Neumann algebra of operators on a separable Hilbert space. When there exists an invariant normal faithful state, the cyclic properties of such channels are investigated passing through the decoherence free algebra and the fixed points domain. Both these spaces are proved to be images of a normal conditional expectation so that their consequent atomic structure are analyzed in order to give a better description of the action of the channel and, for instance, of its Kraus form and invariant densities.Point (ii) already appeared in [14] (see also references therein) and was the original representation of the decoherence free algebra used in [9] when introducing environmental decoherence.The following results are well known.Proposition 4. Assume that there is a faithful normal invariant state for Φ. Then (i) F is a von Neumann subalgebra.(ii) The restriction Φ| N is a *-automorphism.Proof. See e.g.[39] and references therein. Maps with a faithful invariant stateIn this section, we assume that there is a faithful normal state ρ ∈ S(H) for Φ. In this case, there is another special subalgebra investigated in the literature, e.g. [7,27], appearing in some asymptotic splitting, usually called the reversible subalgebra and denoted by M r . We describe the reversible subalgebra, following [7], [27] or [29]. Let S be the closure of the semigroup of channels {Φ n , n ∈ N} in the point-ultraweak topology and defineWe will show below, in Theorem 1, that the equality M r = N holds for channels on B(H) (or more generally on atomic von Neumann algebras).Due to the presence of a faithful normal invariant state, for any ϕ ∈ B(H) * , the set {Φ n * ϕ, n ∈ N} is weakly relatively compact, equivalently, the set S consists of normal operators and is a compact semitopological semigroup ([27, Proposition 2.1]). Further, S contains a minimal ideal M(S) which is a compact topological group. Let F be the unit of this group, then M(S) = F •S and F is a normal conditional expectation preserving the invariant state ρ such that T F = F T for all T ∈ S. Finally, M r is a von Neumann algebra and the minimal ideal M(S) acts as a compact group of *-automorphisms on M r ([7, Theorem 1.2 and Corollary 1.3]). This last fact trivially implies, in particular, that M r ⊆ N and that equality holds in finite dimension, but the infinite dimensional case is quite more delicate and tricky. Now let X ∈ B(H) and let O 0 (X) := {Φ k (X), k ∈ N} be the orbit of X under {Φ k } k≥0 . Then the weak*-closureŌ 0 (X) is the orbit of X under S, O 0 (X) = O(X) := {T (X), T ∈ S} and we can define the stable subspace as M s := {X ∈ B(H), 0 ∈ O(X)}.The following lemma can be deduced from [29, Theorem 2.1], [7, Theorem 1.2] and [27, Proposition 2.2]. Since we did not find an explicit and comprehensive statement in the literature, we reconstruct here the detailed result that we need and the proof for the convenience of the reader.Lemma 1. Let F be the normal conditional expectation introduced before.

Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

“…Here instead, we once again adopt an approach based on functional inequalities for discrete-time QMS. Unfortunately, there is not a lot of work on functional inequalities for non-primitive evolutions in discrete time [9,52], which is particularly important in our setting. To start amending this gap in the literature and increase the class of examples our techniques apply to, we generalize the framework of Poincaré inequalities to discrete-time non-primitive QMS.…”

Absorption in Invariant Domains for Semigroups of Quantum Channels

“…The structure of the set F(P) is well known when there exists a faithful normal invariant state (i.e., the semigroup is positive recurrent): In such a case, it is an atomic W * -algebra, since it is in the multiplicative domain and it is the range of a P-invariant normal conditional expectation (see, for instance, [21,22,29] and more recent developments in [4,7,19]).…”

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“…Various fields of applications to quantum physics motivate the study of these channels and, in particular, of absorption operators and fixed points (see [1,13,32,33,37]), but the approach we shall follow here comes from classical and quantum probability. In a first step, a central theme which stimulated this work is the study of the fixed points of a quantum channel: This theme has been widely investigated, with satisfactory results for the case of positive recurrent channels, i.e., channels with an invariant faithful density (see [5,7,9,10,19,27]), but information is lacking and yet useful in the general case [1,2,24]. Once the construction of the mathematical object clearly appeared, we realized that it could be a useful tool to tackle the different aforementioned subjects.…”

Generalized Dobrushin Coefficients on Banach Spaces

“…Positive (very often completely positive) operators T describing quantum evolutions act on ordered Banach spaces (cf. [1,2,5,6,10,11,29,30,35]). The topic of positive linear operators on ordered Banach spaces, which are not necessarily Banach lattices, has attracted significant attention (e.g., [8,[14][15][16][17][24][25][26][27][28]32])…”