Passage Times, Exit Times and Dirichlet Problems for Open Quantum Walks

Bardet, Ivan; Bernard, Denis; Pautrat, Yan

doi:10.1007/s10955-017-1749-3

Cited by 18 publications

(41 citation statements)

References 35 publications

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“…Let, for any vertices x, y, P n (x, y) := n C∈P(x→y;n)Ĉ where P(x → y; n) denotes the set of products of effect matrices of T associated with all paths of length n moving from vertex x to y, andĈ is the conjugation mapĈρ := CρC † so by going through all paths C = B ini n−1 · · · B i 3 i 2 B i 2 i 1 allowed by the QMC T we obtain the above operator. The fact that such operator is well-defined follows from the reasoning described, for instance, in [8]. and soΩ jj F (1) = U (1)F (1) =⇒Ω jj F (1) =Ẑ jj F jπ ApplyD jj on the left of both sides and, noting that Tr(D jjΩjj ρ) = Tr(ρ) and that (DẐ) jj =D jjẐjj we obtain, after taking the trace, Tr(N jπ ) = Tr[(DẐ) jj F jπ ]…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

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

Lardizabal

2019

Quantum Inf Process

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“…where B ∈ B(K R ) and P j := S * T (R j ⊗ I R )T * S are the minimal central projections in F . Moreover, since E F is given by (5) and satisfies (6), we see that a state ξ is invariant for Φ if and only if it is invariant for E F . Consequently, by Lemma 2 (iv), any state of the form ψ = T * S(ω j ⊗ σ j )S * T with ω j ∈ S(L j ) is an invariant state for Φ.…”

Section: Reducible Mapsmentioning

confidence: 93%

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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“…Among the different notions of recurrence appearing in the quantum literature, we will refer here to a recent one based on a monitoring process, developed for unitary quantum walks [1,9,19,24], and later extended to open quantum walks [5,14,18,26]. Consider a discrete time evolution given by iterating a quantum channel Φ on a Hilbert space H. Given a subspace H 0 ⊂ H, we will identify I(H 0 ) with the subspace constituted by those operators ρ ∈ I(H) with ran ρ ⊂ H 0 and ker ρ ⊃ H ⊥ 0 .…”

Section: Recurrence For Quantum Markov Chains and Schur Functionsmentioning

confidence: 99%

“…We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].This work provides a detailed study of the consequences of the splitting properties for FR-functions regarding recurrence in quantum Markov chains. Two kinds of FR-function splittings, related to factorizations and decompositions into sums of the underlying operator, yield two types of splitting rules for quantum Markov chains and, thus, for the particular case of classical Markov chains.…”

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Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

Grünbaum

Lardizabal

Velázquez

2019

Ann. Henri Poincaré

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In this work we study the recurrence problem for quantum Markov chains, which are quantum versions of classical Markov chains introduced by S. Gudder and described in terms of completely positive maps. A notion of monitored recurrence for quantum Markov chains is examined in association with Schur functions, which codify information on the first return to some given state or subspace. Such objects possess important factorization and decomposition properties which allow us to obtain probabilistic results based solely on those parts of the graph where the dynamics takes place, the so-called splitting rules. These rules also yield an alternative to the folding trick to transform a doubly infinite system into a semi-infinite one which doubles the number of internal degrees of freedom. The generalization of Schur functions -so-called FR-functions-to the general context of closed operators in Banach spaces is the key for the present applications to open quantum systems. An important class of examples included in this setting are the open quantum random walks, as described by S. Attal et al., but we will state results in terms of general completely positive trace preserving maps. We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.walks on graphs, see [32] for a recent survey on the subject. Regarding hitting probabilities and recurrence in the setting of OQWs, see [5,14,26,30].This work provides a detailed study of the consequences of the splitting properties for FR-functions regarding recurrence in quantum Markov chains. Two kinds of FR-function splittings, related to factorizations and decompositions into sums of the underlying operator, yield two types of splitting rules for quantum Markov chains and, thus, for the particular case of classical Markov chains. As a consequence of these splitting rules we will see that, similarly to UQWs, the return probability is also invariant under certain local perturbations.Other novel contributions of this paper deal with generalizations of known results on recurrence in classical Markov chains and UQWs. Different quantum generalizations of Kac's lemma for classical Markov chains are commented in Subsect. 2.1. Besides, Theorem 2.8 constitutes the quantum version of a well known result for classical Markov chains, namely, that finiteness and irreducibility imply the positive recurrence of every state, i.e. every state returns to itself with probability one and in a finite expected return time. A similar result holds for finite-dimensional unitary evolutions, which in addition present the striking particularity of exhibiting integer valued expected times for the return to any state [19]. These results has been generalized to the return to a subspace in [9], while [33] proves that they hold for the return to a state in iterated quantum channels whenever they are unital. We extend this property of unital quantum channels to the r...

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Passage Times, Exit Times and Dirichlet Problems for Open Quantum Walks

Cited by 18 publications

References 35 publications

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

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

On Period, Cycles and Fixed Points of a Quantum Channel

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

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