Irreducible and periodic positive maps

109

“…The following is Theorem 5.4 from [10], which again extends to the infinite dimensional case, with the same proof.…”

Section: Period and Aperiodicity For Oqrwsmentioning

confidence: 73%

Section: Irreducibility For Oqrwsmentioning

confidence: 99%

“…Here we follow Fagnola and Pellicer ( [10]) and Groh ([13]). We define d − to be subtraction modulo d.…”

Section: Period and Aperiodicity For Oqrwsmentioning

confidence: 99%

“…• a notion of period, together with associated results on the peripheral spectrum, that were defined in the same setting by Groh ([13]) and extended by Fagnola and Pellicer ( [10]);…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Carbone

Pautrat

2015

109

“…Afterward, in Section 3, we introduce the study of cycles, using also the notion of period, as introduced in the quantum context in [20], and generalizing it to the infinite dimensional case. We start from the irreducible case, where the relation between the DFA, the fixed points of powers of the channel, and the cyclic decomposition is evident and can be clearly described (Corollary 2 and Proposition 6).…”

Section: Introductionmentioning

confidence: 99%

On Period, Cycles and Fixed Points of a Quantum Channel

Carbone

Jenčová

2019

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.

Landauer’s Principle for Trajectories of Repeated Interaction Systems

Hanson

Joye

Pautrat

et al. 2018

We analyze Landauer's principle for repeated interaction systems consisting of a reference quantum system S in contact with an environment E which is a chain of independent quantum probes. The system S interacts with each probe sequentially, for a given duration, and the Landauer principle relates the energy variation of E and the decrease of entropy of S by the entropy production of the dynamical process. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated to a two-time measurement protocol of, essentially, the energy of E. The emphasis is put on the adiabatic regime where the environment, consisting of T 1 probes, displays variations of order T −1 between the successive probes, and the measurements take place initially and after T interactions. We prove a large deviation principle and a central limit theorem as T → ∞ for the classical random variable describing the entropy production of the process, with respect to the FS measure. In a special case, related to a detailed balance condition, we obtain an explicit limiting distribution of this random variable without rescaling. At the technical level, we obtain a non-unitary adiabatic theorem generalizing that of [HJPR17] and analyze the spectrum of complex deformations of families of irreducible completely positive trace-preserving maps.1 we will always set the Boltzmann constant to 1, so that β = 1/Θ, Θ the temperature.