Open Quantum Random Walks, Quantum Markov Chains and ...

OPEN QUANTUM RANDOM WALKS AND ASSOCIATED QUANTUM MARKOV CHAINS

arXiv:1608.01065v1 [math.FA] 3 Aug 2016

AMEUR DHAHRI AND FARRUKH MUKHAMEDOV Abstract. In the present paper, we establish a connection between Open Quantum Random Walks and the Quantum Markov Chains. In particular, we construct two kinds of Quantum Markov Chains associated with Open Quantum Random Walks. Moreover, we study the recurrence and the property of accessibility associated to these Quantum Markov Chains. Mathematics Subject Classification: 46L35, 46L55, 46A37. Key words: Open quantum random walk; quantum Markov chain; recurrence, accessibility.

1. Introduction Random walks [14] are a useful mathematical concept, which found successful applications in many branches of natural sciences. There are several kinds of approaches to investigate quantum analogies of random walks (see [16, 17]). Very recently, in [8] a new family of quantum random walks, called Open Quantum Random Walks (OQRWs) has been introduced. These random walks deal with density matrices instead of pure states, that is, on a state space H ⊗ K they consider density matrices of the form X ρ= ρi ⊗ |iihi|. i

To this density matrix is attached a probability distribution, associated with the values one would obtain by measuring the position: P rob({i}) = Tr(ρi ).

After one step of the dynamics, the density matrix evolves to another state of the same form X ρ′ = ρ′i ⊗ |iihi|. i

with the associated new distribution. It is proved that these open quantum random walks are a noncommutative extension of all the classical Markov chains, i.e. they contain all the classical Markov chains as particular cases, but they also describe typically quantum behaviors. Furthermore, in [9, 17, 18, 22] long term behavior of these quantum random walks have been studied. Note that OQRWs is defined by a completely positive mapping given by XX X X (1.1) M(ρ) = Bji ρj Bji∗ ⊗ |iihi|, ρ = ρi ⊗ |iihi|, Tr(ρi ) = 1, i

j

i

Bji

i

where the are matrices associated to a transition from site |ji to site |ii, for all i, j [8]. It is P imposed the condition i Bji∗ Bji = I for all j. In recent works [11, 12, 19] it has been studied irreducibility and periodicity aspects of the mapping (1.1). On the other hand, it is known [1, 4] that a Quantum Markov Chain (QMC) is a quantum generalization of a classical Markov Chain where the state space is a Hilbert space, and the transition probability matrix of a Markov chain is replaced by a transition amplitude matrix, 1

2

AMEUR DHAHRI AND FARRUKH MUKHAMEDOV

which describes the mathematical formalism of the discrete time evolution of open quantum systems (see [2, 4, 21] for more details). The quantum analogues of Markov chains were first constructed in [1], where the notion of quantum Markov chain on infinite tensor product algebras was introduced. In [5, 6] it was introduced stopping times for QMC, and studied recurrence and transience in terms of the stopping times. The reader is referred to [3, 10, 13, 20] and the references cited therein, for recent developments of the theory and the applications. In [7] it was studied a construction of QMC associated with unitary quantum random walks. Hence, it is natural to construct and investigate QMC associated with OQRWs. In this paper, we discuss the construction of Quantum Markov Chains from Open Quantum Random Walks, and we examine the notions of recurrence. Moreover, we study the classification of projection for the property of accessibility. Note that in [18] a notion of recurrence is studied for some classes of OQRWs. We also mention that transience and recurrence have been studied in [15] for quantum Markov semigroups. In our paper, we propose to study the recurrence of OQRWs from QMC perspectives. The paper is organized as follows. In section 2, we recall basic concepts related to the Open Quantum Random Walks. We further describe the notions of Quantum Markov Chains and E-recurrence and ϕ-recurrence, respectively, this is given in section 3. Finally, in section 4, we construct two kinds of Quantum Markov Chains associated with Open Quantum Random Walks, and we study the notions of recurrence and accessibility property. 2. Open Quantum Random Walks In this section we recall basic setup to define Open Quantum Random Walks (OQRW). Let K denote a separable Hilbert space and let {|ii}i∈Λ be an orthonormal basis indexed by the vertices of some graph Λ. Here the set Λ of vertices might be finite or countable. The elements of such basis will be called sites (or vertices). Let H be another Hilbert space, which will describe the degrees of freedom given at each point of Λ. Then we will consider the space H ⊗ K. For each pair i, j one associates a bounded operator Bji on H. This operator describes the effect of passing from |ji to |ii. We will assume that for each j, one has X (2.1) Bji∗ Bji = 1I, i

where, if infinite, such series is strongly convergent. This constraint means: the sum of all the effects leaving site j is 1I. The operators Bji act on H only, we dilate them as operators on H ⊗ K by putting Mji = Bji ⊗ |iihj| .

The operator Mji encodes exactly the idea that while passing from |ji to |ii on the lattice, the effect is the operator Bji on H. According to [8] one has X ∗ (2.2) Mji Mji = 1I. i,j

Therefore, the operators (Mji )i,j define a completely positive mapping XX ∗ (2.3) M(ρ) = Mji ρ Mji i

on H ⊗ K.

j


3

In what follows, we consider density matrices on H ⊗ K with the particular form X ρ= ρi ⊗ |iihi|, i

P

assuming that i Tr(ρi ) = 1. For a given initial state of such form, the Open Quantum Random Walk (OQRW) is defined by the mapping M, which has the following form XX (2.4) M(ρ) = Bji ρj Bji∗ ⊗ |iihi|. i

j

Hence, a measurement of the position in K would give that each site i is occupied with probability X ∗ Tr Bji ρj Bji . j

If the measurement is performed after two steps, i.e. XXX ∗ ∗ M2 (ρ) = Bji Bkj ρk Bkj Bji ⊗ |iihi| . i

j

k

Hence measuring the position, we get the site |ii with probability XX ∗ ∗ Tr Bji Bkj ρk Bkj Bji . j

k

The random walk which is described this way by the iteration of the completely positive map M is not a classical random walk, it is a quantum random walk. In this paper, we will show that the above appeared distributions of OQRW will define a quantum Markov chain. This allows us treat such quantum walks in framework of QMC. 3. Quantum Markov Chains and recurrence In this section, we recall a definition of quantum Markov chains [1, 3, 4] and define a new notion called ϕ-recurrence. For each i ∈ Z+ , (here Z+ denotes the set of all non negative integers) let us associate identical copies of a separable Hilbert space H and C ∗ -subalgebra M0 of B(H), where B(H) is the algebra of bounded operators on H : H{i} = H,

(3.1)

A{i} = M0 ⊂ B(H) for each i ∈ Z+

We assume that any minimal projection in M0 is one dimensional. For any bounded Λ ⊂ Z+ , let O AΛ = Ai i∈Λ

(3.2)



A=

[

Λ⊂Z+ ,|Λ| n Remark 3.1. We notice that if the state φ0 satisfies the following condition: (3.6)

φo (E(1I ⊗ x)) = φ0 (x), x ∈ M0

then the Markov pair (φ0 , E) defines local states (3.7)

ϕ[i,n] (xi ⊗ xi+1 ⊗ . . . ⊗ xn ) = φ0 (E(xi ⊗ E(xi+1 ⊗ · · · ⊗ E(xn ⊗ 1I) · · · )))).

The family of local states {ϕ[i,n]}, due to (3.3),(3.6), satisfies a compatibility condition, and N therefore, the state ϕ is well defined on AZ := Ai . Moreover, ϕ is translation invariant, i.e. i∈Z

it is invariant with respect to the shift α, i.e. α(Jn (a)) = Jn+1 (a).

Recall that by Tr we denote the trace on M0 which takes the value 1 at each minimal (i) f be the trace on M0 ⊗ M0 . Denote by f projection, and let Tr Tr , i = 1, 2, the partial traces defined by (3.8)

(1) (2) f Tr (a ⊗ b) = Tr(a)b, f Tr (a ⊗ b) = Tr(b)a

In [21] it was given a construction of a quantum Markov chain defined by a set {Ki }i∈N of conditional density amplitudes [1]. Namely, let W0 ∈ M0 be a density matrix and {Ki }i∈N be a set of the Hilbert-Schmidt operators in M0 ⊗ M0 satisfying X kKi k2 < ∞, i

X

(3.9)

i

f(2) (Ki K ∗ ) = 1I. Tr i

Then the corresponding transition expectation [4] X (2) f (3.10) E(A) = Tr (Ki AKi∗ ), A ∈ M0 ⊗ M0 . i

and the density operator W0 form a Markov pair (W0 , E).


5

Remark 3.2. We point out if additionally W0 satisfies X (1) f (3.11) Tr (Ki∗ (W0 ⊗ 1)Ki ) = W0 . i

then the associated QMC is well defined on the algebra AZ .

Following [5] we recall a definition of the stopping time associated with a projection e ∈ M0 , which is a sequence {τk } defined by τ0 = e ⊗ 1I[1 = J0 (e),

τ1 = e⊥ ⊗ e ⊗ 1I[2 = J0 (e⊥ )J1 (e), ······ , τk = e|⊥ ⊗ ·{z · · ⊗ e⊥} ⊗e ⊗ 1I[k+1 = J0 (e⊥ ) · · · Jk (e⊥ )Jk (e), k

n τ∞

= |e ⊗ ·{z · · ⊗ e⊥} ⊗1I[n+1 = J0 (e⊥ ) · · · Jn (e⊥ ). ⊥

n+1

n Since the sequence {τ∞ } is decreasing, therefore, its strong limit exists in A′′ , and it is denoted by n τ∞ := lim τ∞ n→∞

One can see that

X

(3.12)

k≥0

τk = 1I − τ∞ ,

where the sum is meant in the strong topology in A′′ . Definition 3.3. Let ϕ be a QMC on A associated with the pair (φ0 , E). A projection e is called (i) E-completely accessible if n E0] (τ∞ ) := lim E0] (τ∞ ) = 0; n→∞

(i) ϕ-completely accessible if ϕ(τ∞ ) = 0; (iii) E-recurrent if Tr(E(e ⊗ 1I)) < ∞ and one has X ! 1 Tr E0] J0 (e) ⊗ τk =1 Tr(E(e ⊗ 1I)) n≥0 (iv) ϕ-recurrent if ϕ(J0 (e)) 6= 0 and

X 1 ϕ J0 (e) ⊗ τk ϕ(J0 (e)) n≥0

!

=1

Definition 3.4. Let ϕ be a QMC on A associated with the pair (φ0 , E) and e, f be two projections in M0 . A projection f is called (i) E-accessible from e if there is n ∈ N such that E0] (J0 (e) ⊗ 1In−1] ⊗ Jn (f )) 6= 0;

(ii) ϕ-accessible from e (we denote it as e →ϕ f ) if there is n ∈ N such that ϕ(J0 (e) ⊗ 1In−1] ⊗ Jn (f )) 6= 0.

6


If e →ϕ f and f →ϕ e, then e and f are called ϕ-communicate and one denotes e ↔ϕ f . Remark 3.5. We notice that the E-accessibility, E-recurrence have been introduced and studied in [5]. From the definitions one can infer that, due to the translation-invariance of QMC ϕ, Eaccessibility and E-recurrence imply ϕ-accessibility and ϕ-recurrence, respectively. The reverse is not true (see Example 4.5). Now we are going to study several properties of ϕ-accessibility and ϕ-recurrence, respectively. Theorem 3.6. Let ϕ be a QMC on A associated with the pair (φ0 , E). The following statements hold: (i) ϕ(Jn (e)) = 0 for all n ∈ N if and only if for every k ∈ N one has ϕ(β k (τ∞ )) = 1, where β(a0 ⊗ a1 ⊗ · · · an ) = 1I ⊗ a0 ⊗ a1 ⊗ · · · an , for any n ∈ N;

(ii) e is ϕ-recurrent if and only if ϕ(J0 (e) ⊗ τ∞ ) = 0. In particularly, if e is ϕ-completely accessible, then e is ϕ-recurrent; (iii) if ϕ is faithful, then e is ϕ-completely accessible if and only if e is ϕ-recurrent; (iv) if all projections in M0 ϕ-communicate and e is ϕ-recurrent, then e is ϕ-completely accessible. Proof. (i) Let ϕ(Jn (e)) = 0 for every n ∈ N. For any k, m ∈ N we have 1Im] ⊗ τk ≤ 1Im+k−1] ⊗ Jm+k (e) ⊗ 1I[m+k+1 ,

therefore, one finds ϕ(τk ) ≤ ϕ(Jm+k (e)) = 0. Hence, from (3.12) one gets ϕ(β m (τ∞ )) = 1. Now assume that ϕ(β m (τ∞ )) = 1 any m ∈ N. Then again from (3.12) we obtain X τk = 0, ϕ βm k≥0

m

which implies ϕ(β (τk )) = 0 for all k ∈ N. This means ϕ(β m (τ0 )) = ϕ(Jm (e)) = 0. (ii) Let e be ϕ-recurrent. Then from the definition and (3.12) one finds X τk = ϕ(Jm (e)) − ϕ(Jm (e) ⊗ τ∞ ), ϕ(J0 (e)) = ϕ J0 (e) ⊗ k≥0

which means ϕ(J0 (e) ⊗ τ∞ ) = 0. The reverse implication is obvious. (iii) if ϕ is faithful, then ϕ-completely accessibility is equivalent to τ∞ = 0, then from (ii) we have that e is ϕ-recurrent. Conversely, if e is ϕ-recurrent, then due to the faithfulness of ϕ with (ii) one gets J0 (e) ⊗ τ∞ = 0, so τ∞ = 0 which means ϕ-completely accessibility of e. (iv) Assume that e is not ϕ-completely accessible, this means ϕ(τ∞ ) > 0. Due to ϕ-recurrence one has ϕ(J0 (e) ⊗ τ∞ ) = 0, which implies that n lim ϕ(J0 (e) ⊗ τ∞ ) = 0.

n→∞

The last equality yields that n lim ϕ(J0 (e) ⊗ J1 (e) ⊗ τ∞ ) = 0,

n→∞

n lim ϕ(J0 (e) ⊗ J1 (e⊥ ) ⊗ τ∞ ) = 0,

n→∞

so n lim ϕ(J0 (e) ⊗ 1I ⊗ τ∞ ) = 0.

n→∞

Hence, iterating the last equality, for every k ∈ N one finds (3.13)

ϕ(J0 (e) ⊗ 1Ik−1] ⊗ τ∞ ) = 0.


7

Since ϕ(τ∞ ) > 0, then one can find a projection p ∈ M0 (p 6= 0) such that τ∞ ≥ λp for some positive number λ. Then from (3.13) we infer that 1 ϕ(J0 (e) ⊗ 1Ik−1] ⊗ p) ≤ ϕ(J0 (e) ⊗ 1Ik−1] ⊗ τ∞ ) = 0 λ this implies that e and p are not ϕ-communicate. This is a contradiction. This completes the proof. 4. Quantum Markov Chains associated with OQRW In this section, we are going to construct two kinds of QMCs associated with OQRW. Let M be a OQRW given by (2.4). In this section we will use notations from the previous sections. Take a density operator ρ ∈ B(H ⊗ K), of the form X ρ= ρi ⊗ |iihi|, i

where ρi 6= 0 for all i. We are going to construct a QMC associated with ρ and M. To do so, let us denote Iρ = {i ∈ Λ : ρi = 0}.

By Kρ we denote a subspace of K generated by the basis {|iihi|}i∈Iρ . Put Kρ = K ⊖ Kρ . In what follows, we consider the algebra O Ai , Aρ = i∈Z+

where Ai = B(H ⊗ Kρ ) for all i ∈ Z+ . Now we consider the OQRW associated with {Mji }i∈Λ,j∈Iρc , here Iρc = Λ \ Iρ . From (2.2) we infer that X ∗ Mji Mji = 1I, (4.1) i∈Λ,j∈Iρc

here the identity operator defined on B(H ⊗ Kρ ). Define the following operators: 1 1/2 Aij = (4.2) ρ ⊗ |iihj| , (Tr(ρj ))1/2 j (4.3)

i ∈ Λ, j ∈ Iρc ,

Kij = Mji∗ ⊗ Aij .

Let us show that the pair (ρ, {Kij }) defines a QMC on Aρ . Let us check the condition (3.9) which follows from (4.1). Indeed, one has X X Tr(ρj ⊗ |iihi|) (2) ∗ Tr Kij Kij = Mji∗ Mji Tr(ρj ) i,j i,j (4.4)

= 1I.

Hence, one can define a quantum Markov chain ϕ corresponding to the pair (ρ, E), where the transition expectation E has the following form (see (3.10)): X Tr(ρj ⊗ |jihj|y) E(x ⊗ y) = Mji∗ xMji (4.5) . Tr(ρj ) i,j

8


Now it is interesting to know whether the defined QMC can be extended to the algebra O AρZ := Ai . i∈Z

The next result gives an affirmative answer to this question.

Theorem 4.1. The QMC ϕ associated with the Markov pair (ρ, E) can be extended to AρZ .

Proof. The extension of ϕ on AZ is defined by (3.7). It is compatible, if the condition (3.11) is satisfied. Therefore, we check (3.11) for (4.3). Indeed, we have X X (1) (1) i ∗ i∗ ∗ Mj ⊗ Aij (ρ ⊗ 1I)Mj ⊗ Aij Kij (ρ ⊗ 1I)Kij = Tr Tr i,j

i,j

=

X

Tr(1)

i,j

=

(4.6)

X

Mji ρMji∗ ⊗ A∗ij Aij

Tr Mji ρMji∗

i,j

Now using Tr Mji ρMji∗

ρj ⊗ |jihj| . Tr(ρj )

= Tr Bji ⊗ |iihj|ρBji∗ |jihi| X Tr Bji ⊗ |iihj|ρℓ ⊗ |ℓihℓ|Bji∗|jihi| = ℓ

(4.7)

= Tr Bji ρj Bji∗ |iihi| = Tr Bji ρj Bji∗ = Tr Bji∗ Bji ρj

from (4.7) one finds X X ρj ⊗ |jihj| (1) ∗ Kij (ρ ⊗ 1I)Kij Tr Bji∗ Bji ρj = Tr Tr(ρj ) i,j i,j =

X j

=

X

Tr

X i

Tr(ρj )

j

=

X j

(4.8)

Bji∗ Bji ρj

ρj ⊗ |jihj| Tr(ρj )

ρj ⊗ |jihj| Tr(ρj )

ρj ⊗ |jihj|

= ρ.

Hence, the above defined QMC can be extended to AρZ .

Theorem 4.2. The QMC ϕ on Aρ has the following form: X (4.9) ϕ(x1 ⊗ x2 ⊗ · · · ⊗ xn ) = Tr(ρv )ψv (x1 )ψv (x1 )ψv (x2 ) · · · ψv (xn ), v


where (4.10)

ψv (x) =

9

1 X Tr Bvi ρv (Bvi )∗ ⊗ |iihi|x , v ∈ Iρc . Tr(ρv ) i

To prove the theorem we need the following auxiliary fact. P Lemma 4.3. Let ρ = ℓ ρℓ ⊗ |ℓihℓ|. Then one has: (i) Tr(ρMvu∗ xMvu ) = Tr(Bvu ρv Bvu∗ ⊗ |uihu|x);

(ii) Tr(ρv ⊗ |vihv|Mji∗xMji ) = Tr(Bji ρv Bji∗ ⊗ |iihi|x)δvj . Proof. (i). We have Tr(ρMvu∗ xMvu ) =

X ℓ

=

X ℓ

=

X ℓ

Tr(ρℓ ⊗ |ℓihℓ|Mvu∗ xMvu ) Tr Bvu ⊗ |uihv|(ρℓ ⊗ |ℓihℓ|)(Bvu∗ ⊗ |vihu|x

Tr Bvu ⊗ |uihv|(ρℓ ⊗ |ℓihℓ|)(Bvu∗ ⊗ |vihu|x

= Tr(Bvu ρv Bvu∗ ⊗ |uihu|x).

The equality (ii) can be proven by the same manner.

Proof of Theorem 4.2. It is enough to prove for the case n = 2 since the general formula can be proved by the induction. From (4.5) taking into account Lemma 4.3 one finds ϕ(x1 ⊗ x2 ) = Tr(ρE(x1 ⊗ E(x2 ⊗ 1I))) X ! = Tr ρE(x1 ⊗ Mji∗ x2 Mji i,j

=

X i,j

=

X ! Tr(ρv ⊗ |vihv|Mji∗ x2 Mji ) Tr ρ Mvu∗ x1 Mvu Tr(ρv ) u,v

X

Tr ρMvu∗ x1 Mvu

X

X Tr Bvi ρv Bvi∗ ⊗ |iihi|x2 Tr Bvu ρv Bvu∗ ⊗ |uihu|x1 Tr(ρv ) i

u,v

=

u,v

=

X i,j

=

Tr ρE(x1 ⊗ Mji∗ x2 Mji )

X

X Tr(ρv ⊗ |vihv|Mji∗ x2 Mji ) Tr(ρv ) i,j

Tr(ρv )ψv (x1 )ψv (x2 )

v

This completes the proof.

Using the same idea of the proof we can get the following

10


Corollary 4.4. For any projection e ∈ B(H ⊗ Kρ ) one has X n+1 n E(e ⊗ E0] (τ∞ )) = Mvu∗ eMvu ψv (e⊥ ) u,v

Example 4.5. Let us consider a stationary OQRW on Z with nearest-neighbor jumps (see [8]). Let H be a Hilbert space and B, C ∈ B(H) such that B ∗ B + C ∗ C = 1I. We define the walk as follows: assume that Bii−1 = B and Bii+1 = C for all i ∈ Z, all the others Bji being equal to 0. Then one can calculate that X (4.11) M(ρ) = B ⊗ |j − 1ihj|ρB ∗ ⊗ |jihj − 1| + C ⊗ |j + 1ihj|ρC ∗ ⊗ |j + 1ihj| . j

Take a density operator ρ ∈ B(H ⊗ K), of the form X ρ= ρi ⊗ |iihi|, i

with ρi 6= 0 for all i. Take a projection e = q ⊗ |kihk| for some k ∈ Z and a projection p ∈ B(H). Then due to Corollary 4.4 and (4.11) we have X n n B ∗ ⊗ |ℓihℓ − 1|eB ⊗ |ℓ − 1ihℓ| ψℓ (e⊥ ) E(e ⊗ E0] (τ∞ )) = ℓ

+

X ℓ

C ∗ ⊗ |ℓihℓ + 1|eB ⊗ |ℓ + 1ihℓ| ψℓ (e⊥ )

= B ∗ qB ⊗ |k + 1ihk + 1| ψk+1 (e⊥ )

n

+C ∗ qC ⊗ |k − 1ihk − 1| ψk−1 (e⊥ )

(4.12)

Taking into account (4.10) one finds   1− ψℓ (e⊥ ) = (4.13)  1−

Tr(Bρk+1 B ∗ q) , if Tr(ρk+1 )

ℓ=k+1

Tr(Cρk−1 C ∗ q) , if Tr(ρk−1 )

ℓ=k−1

n

n

Therefore, from (4.12) with (4.13) one gets E(e ⊗

n E0] (τ∞ ))

(4.14)

n Tr(Bρk+1 B ∗ q) = B qB ⊗ |k + 1ihk + 1| 1 − Tr(ρk+1 ) n Tr(Cρk−1 C ∗ q) ∗ . +C qC ⊗ |k − 1ihk − 1| 1 − Tr(ρk−1 ) ∗

Hence, if one has (4.15)

Tr(Bρk+1 B ∗ q) < Tr(ρk+1 ), Tr(Cρk−1 C ∗ q) < Tr(ρk−1 )

n then from (4.14) we infer that E(e ⊗ E0] (τ∞ )) → 0 as n → ∞ which according to [5, Theorem 1, (iii)] implies that e is E-recurrent. Now let us look for the ϕ-recurrence. From (4.9) we have X n ϕ(e ⊗ |e⊥ ⊗ ·{z · · ⊗ e⊥}) = Tr(ρℓ )ψℓ (e) ψℓ (e⊥ ) n

ℓ


From ψℓ (e) = with (4.13) we obtain ϕ(e ⊗

n τ∞ )

(4.16)

  

Tr(Bρk+1 B ∗ q) , if Tr(ρk+1 )

ℓ=k+1

Tr(Cρk−1 C ∗ q) , if Tr(ρk−1 )

ℓ=k−1

11

n Tr(Bρk+1 B ∗ q) = Tr(Bρk+1 B q) 1 − Tr(ρk+1 ) n Tr(Cρk−1 C ∗ q) ∗ + Tr(Cρk−1 C q) 1 − . Tr(ρk−1 ) ∗

Clearly, if (4.15) is satisfied then e is ϕ-recurrent. This means that under the condition (4.15) the E-recurrence is equivalent to the ϕ-recurrence. If one of the following conditions (a) supp(ρk+1 )supp(B) = 0 and Tr(Cρk−1 C ∗ q) < Tr(ρk−1 ), (b) supp(ρk−1 )supp(C) = 0 and Tr(Bρk+1 B ∗ q) < Tr(ρk+1 ), (c) supp(ρk+1 )supp(B) = 0 and supp(ρk−1 )supp(C) = 0, is satisfied, then from (4.16) one gets that e is still ϕ-recurrent, while it is not E-recurrent. Now we define another kind of transition expectation E˜ : B(H⊗Kρ )⊗B(H⊗Kρ ) → B(H⊗Kρ ) related to (4.5) as follows: ˜ ⊗ y) = E(x

(4.17) In particular, we have

X Tr(ρj ⊗ |jihj|x) (Mji )∗ yMji . Tr(ρj ) i,j

˜ I ⊗ y) = M∗ (y). E(1 ˜ also defines another kind of QMC ϕ˜ on Aρ . Hence, the Markov pair (ρ, E)

Theorem 4.6. If ρ is invariant state of M (i.e. M(ρ) = ρ), then the QMC ϕ˜ can be extended to AρZ . Moreover, ϕ˜ is translation invariant. ˜ From (4.17) we have Proof. It is enough to check the equality (3.6) for the pair (ρ, E). ˜ I ⊗ x)) = Tr(ρM∗ (x)) = Tr(M(ρ)x) = Tr(ρx) = ϕ0 (x). ϕ0 (E(1

Correspondingly, the equality

˜ 1 ⊗ E(x ˜ 2 ⊗ · · · ⊗ E(x ˜ n ⊗ 1I)) · · · )) ϕ(x ˜ 1 ⊗ x2 ⊗ · · · ⊗ xn ) = Tr(ρE(x

defines a QMC on AρZ .

Remark 4.7. The proved theorem provides only sufficient condition for the extendability of ϕ˜ to AρZ . Below, we provide (see Corollary 4.11) where without the invariance of ρ the state also extendable to AρZ . P Remark 4.8. We note that if ρ = ρi ⊗ |iihi| is an invariant density operator w.r.t. M, then one has (4.18)

i

X j

Bji ρj Bji∗ = ρi ,

∀ i ∈ Λ.

12


Multiply the above equation by Bii1 and Bii1 ∗ on the left and right, respectively, and summing over i we get X X (4.19) Bii1 Bji ρj Bji∗ Bii1 ∗ = Bii1 ρi Bii1 ∗ = ρi1 , ∀ i ∈ Λ. i,j

i

By induction, we can establish that if ρ is invariant, then one has X n n∗ Biin Biin−1 · · · Bii12 ρi1 Bii12 ∗ · · · Biin−1 Bii∗n = ρi , ∀ i ∈ Λ. (4.20) i1 ,i2 ,...,in

The existence of invariant density operators for OQRW M has been studied in [18]. Lemma 4.9. One has ˜ 1 ⊗ E(x ˜ 2 ⊗ · · · ⊗ E(x ˜ n ⊗ 1I)) · · · ) = E(x

X

i1 ,i2 ,...,in

(4.21)

n∗ n · · · Bii23 Bii12 ⊗ |i1 ihi1 | Bii12 ∗ Bii23 ∗ · · · Biin−1 Biin−1

×ϕi1 (x1 ) · · · ϕin (xn ),

where ϕk (x) =

(4.22)

Tr(ρk ⊗ |kihk|x) . Tr(ρk )

Proof. Let us prove for n = 2. Then from (4.17) with (4.22) we obtain X ! ˜ 1 ⊗ E(x ˜ 2 ⊗ 1I)) = E˜ x1 ⊗ E(x Mji∗ Mji ϕj (x2 ) i,j

=

X i,j

=

X j

=

E˜ x1 ⊗ (Bji∗ Bji ⊗ |jihj|) ϕj (x2 )

E˜ (x1 ⊗ (1I ⊗ |jihj|)) ϕj (x2 )

XX j

u,v

Mvu∗ (1I ⊗ |jihj|)Mvu∗ ϕv (x1 )ϕj (x2 )

X = (Bvj∗ Bvj∗ ⊗ |vihv|)ϕv (x1 )ϕj (x2 ) j,v

which shows that (4.21) is true at n = 2. General setting can be proved by the same argument. Corollary 4.10. The QMC ϕ˜ on Aρ has the following form X i2 i1 i1 ∗ i2 ∗ in in Tr Bin−1 · · · Bi1 Bi0 ρi0 Bi0 Bi1 · · · Bin−1 ϕi0 (x0 ) · · · ϕin (xn ), ϕ(x ˜ 0 ⊗ x1 ⊗ · · · ⊗ xn ) = i0 ,i1 ,...,in

Corollary 4.11. Let ρ = ρ0 ⊗ |ℓihℓ| for some ℓ ∈ Λ. Then the corresponding QMC ϕ˜ on Aρ has the following form (4.23)

ϕ(x ˜ 0 ⊗ x1 ⊗ · · · ⊗ xn ) = ϕ0 (x0 ) · · · ϕ0 (xn ),

where ϕ0 (x) = Tr(ρ0 ⊗ |ℓihℓ|x). Moreover, ϕ˜ is extendable to AρZ by the same formula (4.23).


13

Example 4.12. We consider the example given in section 12.1 of [8]. In our notation this example is given by Λ = {1, 2}, H = C2 (with canonical basis (e1 , e2 )) and transitions are given by √ a 0 1 0 0 p c 0 1 2 1 2 √ B1 = B2 = B2 = B1 = 0 0 0 b 0 q 0 d

where q = 1 − p ∈ (0, 1), |a|2 + |b|2 = |c|2 + |d|2 = 1, 0 < |a|2 , |c|2 < 1. 1. Denote 1 0 ρ˜ = ⊗ |2ih2| =: ρ0 ⊗ |2ih2|. 0 0 One can calculate that 2 X M(˜ ρ) = Bji ⊗ |iihj| (ρ0 ⊗ |2ih2| Bji∗ ⊗ |jihi| i,j=1

=

2 X i=1

B2i ρ0 B2i∗ ⊗ |iihi|.

Due to B21 ρ0 B21∗ = 0, B22 ρ0 B22∗ = ρ0 , we conclude that ρ˜ is an invariant state for M. Hence, the transition expectation (4.17) corresponding to ρ˜ has the following form ˜ ⊗ y) = ϕ0 (x) E(x

(4.24)

2 X

M2i∗ yM2i ,

i=1

where ϕ0 (x) = Tr(ρ0 ⊗ |2ih2|x). Take any projection e ∈ M0 . Then from (4.24) and (3.5) one can calculate that n E0] (τ∞ ) = (ϕ0 (e⊥ ))n+1 1I ⊗ |2ih2|,

n E(e ⊗ E0] (τ∞ )) = ϕ0 (e)(ϕ0 (e⊥ ))n+1 1I ⊗ |2ih2|.

Hence, we conclude that e is E-completely accessible iff E-recurrent iff ϕ0 (e⊥ ) < 1.

(4.25)

n Similarly, due to Corollary 4.11, we have ϕ(τ ˜ ∞ ) = (ϕ0 (e⊥ ))n+1 . Consequently, we infer that e is E-recurrent iff ϕ-recurrent. ˜

2. Now consider an other initial state 1 1 ρ = ρ0 ⊗ |1ih1| + ρ0 ⊗ |2ih2|. 2 2 2 2∗ We assume that c = 0. One can see that B1 ρ0 B1 = 0, B11 ρ0 B11∗ = a2 ρ0 . Hence, we conclude that n n Biin−1 · · · Bii12 Bii01 ρ0 Bii01 ∗ Bii12 ∗ · · · Biin−1 = 0, if there is k0 ∈ {0, . . . , n} such that ik0 6= ik0 −1 . Assume that e⊥ = p ⊗ |1ih1|. Then from (4.22) one finds Hence, we have n ϕ(τ ˜ ∞ ) =

ϕk (e⊥ ) = Tr(ρ0 p)δk1 , k ∈ {1, 2}.

1 X n n Tr Biin−1 · · · Bii12 Bii01 ρ0 Bii01 ∗ Bii12 ∗ · · · Biin−1 ϕi0 (e⊥ ) · · · ϕin (e⊥ ) 2 i ,i ,...,i 0 1

=

n

1 2n a (Tr(ρ0 p))n . 2

14


So, if either |a| < 1 or |a| = 1 and Tr(ρ0 p) < 1 then e is ϕ-completely ˜ accessible. Similarly, one gets 1 n ϕ(J ˜ 0 (e) ⊗ τ∞ ) = a2n (1 − Tr(ρ0 p))(Tr(ρ0 p))n . 2 Hence, we infer that if Tr(ρ0 p) 6= 1, then e is ϕ-completely ˜ accessible iff ϕ-recurrent. ˜ Otherwise, e is ϕ-recurrent, ˜ but not ϕ-complete ˜ accessible if |a| = 1. Now using (4.21) we infer that 2n |a| 0 n ⊗ |1ih1|(Tr(ρ0 p))n E0] (τ∞ ) = 0 |b|2n

hence one concludes that e is E-completely accessible iff ϕ-completely ˜ accessible. By the same argument, we find that e is E-recurrent iff ϕ-recurrent. ˜ Now assume that e = p ⊗ |1ih1| and a = 1. Then from (4.22) one finds ϕk (e⊥ ) = 1 − Tr(ρ0 p)δk1 , k ∈ {1, 2}.

Hence, we have n ϕ(τ ˜ ∞ )

1 X i2 i1 i1 ∗ i2 ∗ in in = Tr Bin−1 · · · Bi1 Bi0 ρ0 Bi0 Bi1 · · · Bin−1 ϕi0 (e⊥ ) · · · ϕin (e⊥ ) 2 i ,i ,...,i 0 1

n

1 (1 − Tr(ρ0 p))n + 1 , 2 this means that ϕ(τ ˜ ∞ ) = 1/2, so e is not ϕ-completely ˜ accessible. =

Acknowledgments A. Dhahri acknowledges support by the NRF grant. References [1] L. Accardi, On noncommutative Markov property, Funct. Anal. Appl. 8 (1975), 1–8. [2] L. Accardi, Local Perturbations of Conditional Expectations, J. Math. Anal. Appl. 72(1979), 34–69. [3] L. Accardi, F. Fidaleo, F. Mukhamedov, Markov states and chains on the CAR algebra, Inf. Dim. Analysis, Quantum Probab. Related Topics 10 (2007), 165–183. [4] L. Accardi, A. Frigerio, Markovian cocycles, Proc. Royal Irish Acad. 83A (1983) 251-263. [5] L. Accardi, D. Koroliuk. Stopping times for quantum Markov chains, Journ. Theor. Probab. Vol. 5, no. 3, pp 521-535, 1992. [6] L. Accardi, D. Koroliuk. Quantum Markov chains: The recurrence problem. In book: Quantum Prob. and Related Topics VII, Springer, Berlin, 1991, 6373. [7] L. Accardi, G.S. Watson, Quantum random walks, In book: Quantum Prob. and Related Topics IV, Springer, Berlin, 1987, pp. 7388. [8] S. Attal, F. Petruccione, C. Sabot, I. Sinayskiy. Open Quantum Random Walks. J. Stat. Phys. 147 (2012), 832–852. [9] S. Attal, N. Guillotin-Plantard, C. Sabot. Central Limit Theorems for Open Quantum Random Walks and Quantum Measurement Records, Ann. Henri Poincar´e 16 (2015), 15-43. [10] P. Biane, L. Bouten, F. Cipriani, N. Konno, N. Privault, Q. Xu. Quantum Potential Theory, Lecture Notes in Mathematics, 1954 Springer-Verlag Berlin Heidelberg, 2008. [11] R. Carbone, Y. Pautrat. Homogeneous open quantum random walks on a lattice, J. Stat. Phys. 160(2015), 1125-1153. [12] R. Carbone, Y. Pautrat. Open quantum random walks: reducibility, period, ergodic properties. Ann. Henri Poincar´e 17(2016), 99-135. [13] M. Fannes, B. Nachtergaele, R.F. Werner, Finitely correlated states of quantum spin chains, Commun. Math. Phys. 144 (1992), 443–490.


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[14] W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York, 1968. [15] F. Fagnola, R. Rebolledo. Transience and recurrence of quantum Markov semigroups. Probab. Theory Relat. Fields 126 (2003), 289306. [16] J. Kempe, Quantum randm walks: an introductory overview, Contemp. Phys. 44(2003) 307-327. [17] N. Konno, H. J. Yoo. Limit Theorems for Open Quantum Random Walks. J. Stat. Phys. 150(2013), 299-319. [18] C. F. Lardizabal, R. R. Souza, On a class of quantum channels, open random walks and recurrence, J. Stat. Phys. 159 (2015), 772-796. [19] C. F. Lardizabal, R. R. Souza, Open Quantum Random Walks: Ergodicity, Hitting Times, Gamblers Ruin and Potential Theory, J. Stat. Phys. DOI 10.1007/s10955-016-1578-9 [20] P.-A. Meyer, Quantum probability for probabilists. Lecture notes in mathematics 1538, Springer, Berlin 1993. [21] Y.M. Park, H.H. Shin, Dynamical entropy of generalized quantum Markov chains over infinite dimensional algebras, J. Math. Phys. 38 (1997), 6287–6303. [22] C. Pellegrini. Continuous Time Open Quantum Random Walks and Non-Markovian Lindblad Master Equations. J. Stat. Phys. 154(2014), 838-865. Ameur Dhahri, Department of Mathematics, Chungbuk National University, Chungdae-ro, Seowon-gu, Cheongju, Chungbuk 362-763, Korea E-mail address: [email protected] Farrukh Mukhamedov, Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia E-mail address: [email protected], farrukh [email protected]