Homogeneous Open Quantum Random Walks on a Lattice

Carbone, Raffaella; Pautrat, Yan

doi:10.1007/s10955-015-1261-6

Cited by 42 publications

(82 citation statements)

References 33 publications

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“…We conclude with section 9, which is dedicated to examples and applications. We start a study of translation-invariant open quantum random walks on Z d continued in [5], and extending that of [2]. We study examples which illustrate our most practical convergence results, namely Corollaries 5.2, 5.4, and 5.6, as well as our decomposition result, Theorem 7.13.…”

Section: Introductionmentioning

confidence: 99%

Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Carbone

Pautrat

2015

Ann. Henri Poincaré

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109

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Section: Introductionmentioning

confidence: 99%

Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Carbone

Pautrat

2015

Ann. Henri Poincaré

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109

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“…The diverse dynamical behaviour of OQWs has been extensively studied [10][11][12][16][17][18][19][20][21]. The asymptotic analysis of OQWs leads to a central limit theorem [22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Microscopic derivation of open quantum walks

Sinayskiy

Petruccione

2015

Phys. Rev. A

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Open Quantum Walks (OQWs) are exclusively driven by dissipation and are formulated as completely positive trace preserving (CPTP) maps on underlying graphs. The microscopic derivation of discrete and continuous in time OQWs is presented. It is assumed that connected nodes are weakly interacting via a common bath. The resulting reduced master equation of the quantum walker on the lattice is in the generalised master equation form. The time discretisation of the generalised master equation leads to the OQWs formalism. The explicit form of the transition operators establishes a connection between dynamical properties of the OQWs and thermodynamical characteristics of the environment. The derivation is demonstrated for the examples of the OQW on a circle of nodes and on a finite chain of nodes. For both examples a transition between diffusive and ballistic quantum trajectories is observed and found to be related to the temperature of the bath.

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“…As we already mentioned in the Introduction, the results of this section include sometimes a revision and improvements or generalizations of different previous studies. The structure of the fixed points domain has already been investigated and one can find various papers in last two decades, see for instance [1,8,10,12,30] and references therein. For the structure of the DFA, there is some interest growing from different fields and we could improve its description in Theorem 2.…”

Section: Reducible Mapsmentioning

confidence: 99%

“…This family of quantum channels has recently become quite popular and have been extensively studied (see [5,12,22,28,33,35]). Here we want to investigate the structure of the DFA associated with an OQRW: we obtain some results in the general case and then expound some particular remarkable classes.…”

Section: Application To Open Quantum Walksmentioning

confidence: 99%

On Period, Cycles and Fixed Points of a Quantum Channel

Carbone

Jenčová

2019

Ann. Henri Poincaré

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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.

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Homogeneous Open Quantum Random Walks on a Lattice

Cited by 42 publications

References 33 publications

Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Open Quantum Random Walks: Reducibility, Period, Ergodic Properties

Microscopic derivation of open quantum walks

On Period, Cycles and Fixed Points of a Quantum Channel

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