Open quantum random walks, quantum Markov chains and recurrence

Dhahri, Ameur; Mukhamedov, Farrukh

doi:10.1142/s0129055x1950020x

Cited by 27 publications

(29 citation statements)

References 36 publications

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“…In this section we briefly recall the definitions of quantum Markov chains [6,7,18,23] and (ir)reducibility [6,7].…”

Section: Quantum Markov Chainsmentioning

confidence: 99%

“…In this section, we construct QMCs associated with OQRWs. As mentioned in the Introduction, this is a slight modification of the one developed in [18]. We will construct a nonhomogeneous QMC, but in [18], a homogeneous QMC was considered.…”

Section: Quantum Markov Chains Associated With Oqrwsmentioning

confidence: 99%

“…Recently the dynamical behavior of OQRWs drew many interests and some works have been done for the ergodicity, hitting times, recurrence, reducibility, etc, of OQRWs [17,18,21,22]. In [18], Dhahri and Mukhamedov constructed the QMCs for the OQRWs and investigated recurrence and accessibility of the QMC. On the other hand the QMC was introduced by Accardi [1,2,3] and further developed [6,7], and has found several applications.…”

Section: Introductionmentioning

confidence: 99%

“…Accardi and Koroliuk, after defining the QMC, developed the quantum versions of reducibility and irreducibility, accessibility, recurrence and transience [6,7]. In this paper we adopt the construction of QMCs for OQRWs done in [18] with some modifications. A remarkable point in our construction is that we have introduced the sub-Markovian transition expectations, contrasting to the fact that it is generally required to have Markovianity for the transition expectations.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Markov Chains Associated with Open Quantum Random Walks

Dhahri

Yoo

2019

J Stat Phys

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In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

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“…In this section we briefly recall the definitions of quantum Markov chains [6,7,18,23] and (ir)reducibility [6,7].…”

Section: Quantum Markov Chainsmentioning

confidence: 99%

Section: Quantum Markov Chains Associated With Oqrwsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum Markov Chains Associated with Open Quantum Random Walks

Dhahri

Yoo

2019

J Stat Phys

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“…Other topics on OQWs which have been examined are the following: reducibility, periodicity, ergodic properties [13]; large deviations [14]; open quantum Brownian motions [44]; site (vertex) recurrence of OQWs [8,15,16,33]. We refer the reader to [45] for a recent survey.…”

Section: Quantum Markov Chains On a Finite Graph Consider The Setmentioning

confidence: 99%

Mean hitting times of quantum Markov chains in terms of generalized inverses

Lardizabal

2019

Quantum Inf Process

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We study quantum Markov chains on graphs, described by completely positive maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008) and which includes the dynamics given by open quantum random walks as defined by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such structures we examine a quantum notion of mean time of first visit to a chosen vertex. However, instead of making direct use of the definition as it is usually done, we focus on expressions for such quantity in terms of generalized inverses associated with the walk and most particularly the socalled fundamental matrix. Such objects are in close analogy with the theory of Markov chains and the methods described here allow us to calculate examples that illustrate similarities and differences between the quantum and classical settings. arXiv:1907.01313v2 [math-ph] 9 Jul 2019 Appendix: proofsThroughout the proofs we recall the notation |e I = |e I n k as n, k are clear from context.Proof of Proposition 6.3. The proof is inspired by [24]. We recall that if M ij denotes the (i, j)-th minor of a matrix A, we define the adjugate matrix as adj(A) := ((−1) i+j M ji ) 1≤i,j≤n , and it holds that A adj(A) = adj(A)A = det(A)I By Sylvester's determinant theorem, we have det(I m + AB) = det(I n + BA), from which we obtain as a consequence that det(X + |c r|) = det(X) + r| adj(X)|c

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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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Open quantum random walks, quantum Markov chains and recurrence

Cited by 27 publications

References 36 publications

Quantum Markov Chains Associated with Open Quantum Random Walks

Quantum Markov Chains Associated with Open Quantum Random Walks

Mean hitting times of quantum Markov chains in terms of generalized inverses

On Period, Cycles and Fixed Points of a Quantum Channel

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