# On Limiting Distributions of Quantum Markov Chains

Authors: Chaobin Liu; Nelson Petulante
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

In a quantum Markov chain, the temporal succession of states
is modeled by the repeated action of a “bistochastic quantum operation” on
the density matrix of a quantum system. Based on this conceptual framework,
we derive some new results concerning the evolution of a quantum system,
including its long-term behavior. Among our findings is the fact that
the Cesàro limit of any quantum Markov chain always exists and equals
the orthogonal projection of the initial state upon the eigenspace of the
unit eigenvalue of the bistochastic quantum operation. Moreover, if the
unit eigenvalue is the only eigenvalue on the unit circle, then the quantum
Markov chain converges in the conventional sense to the said orthogonal
projection. As a corollary, we offer a new derivation of the classic result
describing limiting distributions of unitary quantum walks on finite graphs
(Aharonov et al., 2001).


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## Body

## 1. Introduction

The theory of Markov chains, when appropriately generalized, provides a potent paradigm for analyzing the stochastic evolution of quantum systems. Over the past decade, motivated largely by the prospect of superefficient algorithms, the theory of so-calledquantum Markov chains, especially in the guise of quantum walks, has generated a huge volume of research, including many discoveries of fundamental importance, such as those in [1–10] and numerous recent advances, such as those in [11–15].In the context of quantum walks, the itinerary of the walker is confined to a particular topological network. The walker's every move, from node to adjacent node, is governed by a set of local rules. When applied repeatedly to a given initial state of the system (represented by a superposition of basis states), these rules yield a succession of new states, reflecting, ad infinitum, the evolution of the system. A transition rule can be either unitary (closed) or nonunitary (open), depending, respectively, on whether it is intrinsic to the system or exposes the system to external influences such as decoherence, noise, or measurement.In this paper, we adopt the formalism of “quantum operations,” whereby both unitary and nonunitary rules of state transition, as well as various combinations thereof, are treated under a unified mathematical model. In this framework, the “transition matrix” of a classical Markov chain is replaced by a “bistochastic quantum operation,” and the “state distribution vector” of the classical Markov chain is replaced by a “density matrix.” The resulting description of quantum state evolution, known as aquantum Markov chain [1, 16], turns out to resemble very closely the evolution of a classical Markov chain. In particular, the same kind of questions pertain, including questions about the long-term evolution of the process and the possible existence of a limiting state.Among our findings is the fact that the Cesàro limit of any quantum Markov chain converges always to a stationary “state,” regardless of the initial state. As a noteworthy special case of this result, we remark that for any unitary quantum walk on a graph, as in [1], the limit of the time-averaged probability distribution always exists.To complete the picture, we specify conditions for the existence of a limiting state in the strict (non-Cesàro) sense of the word “limiting.” In the strict sense, it turns out that the limiting behavior depends only on the deployment on the unit disc of the eigenvalues of the bistochastic quantum operation. Specifically, ifλ=1 is the only eigenvalue on the unit circle, then, for any given initial state, the associated quantum Markov chain converges to a stationary state. Moreover, if the eigenspace of λ=1 is one-dimensional or contains only a single density operator, then the associated Markov chain converges to the maximally mixed state, irrespective of the initial state. Otherwise, if the bistochastic quantum operation possesses any eigenvalue on the unit circle other than λ=1, then a limiting state is not guaranteed to exist except in the generalized sense of Cesàro. These findings are seen to be analogous, in a very natural way, to the fundamental properties of classical Markov chains (e.g., in [17, Chapter 8]).Our results may represent substantial progress toward answering the first of a set of “open questions” posed by Ambainis [16]. We would be remiss not to acknowledge our indebtedness to [18], which treats the special case of a quantum Markov chain generated by a random unitary operation.In what follows, after a brief review of some preliminaries (Section2), we proceed (Section 3) to present our main results on the question of the limiting behavior of a quantum Markov chain. The proofs of the results given in Section 3 are deferred to Section 4. In Section 5, we present a comprehensive classification of quantum operations according to their limiting behavior. Finally, in Section 6, we offer some concluding remarks, including some relevant questions for further investigation.


## 2. Bistochastic Quantum Operations

Given a Hilbert spaceℋ of finite dimension N, let 𝔅(ℋ) denote the set of all linear operators on ℋ, with inner product defined by (2.1)〈X,Y〉≡tr⁡(X†Y).The corresponding norm, calledFrobenius norm or Schatten 2-norm, is defined by (2.2)‖X‖≡[tr⁡(X†X)]1/2=〈X,X〉.
This choice of norm on 𝔅(ℋ) will remain in effect throughout this paper.Let𝔇(ℋ)⊂𝔅(ℋ) denote the set of positive operators ρ:ℋ→ℋ with Tr(ρ)=1. The operators ρ∈𝔇(ℋ) are the so-called “density operators.” They serve to model, as faithfully as do the “state vectors” themselves, the possible states of a quantum system whose state vectors reside in ℋ.By asuperoperatorΦ on 𝔅(ℋ), we mean a linear mapping Φ:𝔅(ℋ)→𝔅(ℋ), with norm defined by (2.3)‖Φ‖≡SupX∈B(H)‖Φ(X)‖‖X‖.Note thatdim𝔅(ℋ)=N2, where N=dim(ℋ). Thus, any superoperator Φ on 𝔅(ℋ) can be represented, relative to a given basis for 𝔅(ℋ), by an N2×N2 matrix. In the sequel, this matrix will be denoted by the symbol [Φ]. In particular, relative to a special basis consisting of eigenvectors and generalized eigenvectors of Φ, the shape of the matrix [Φ] conforms to a special quasidiagonal lay-out called the Jordan canonical form. The details can be found in any one of a number of sources, including [18].Among the set of superoperators, we distinguish a special subset called “quantum operations”. By definition, to qualify as a quantum operation, the superoperatorΦ must be completely positive, meaning that the extended map Φ⊗𝕀n is positive for all n≥1.The formalism of quantum operations is flexible enough to handle both unitary (closed) and nonunitary (open), or a mixture thereof, of discrete transitions of state of a quantum system. For a good introductory exposition of this subject, see [19, 20].By Choi's Theorem  [19–22], any completely positive linear operator, including any quantum operation Φ:𝔅(ℋ)→𝔅(ℋ), can be represented in terms of a set 𝒜={Ai∣i=1,2,…,N2} of “Kraus operators” as follows:(2.4)ΦA(X)=∑iAiXAi†.
In this expression, which we call the “Choi expansion” of Φ, the symbol Ai† denotes A̅iT (transpose of the complex conjugate of Ai).In terms of the Choi expansion, the condition of beingtrace preserving, meaning that Tr(Φ𝒜(X))=Tr(X) for all X∈𝔅(ℋ), is equivalent to the condition(2.5)∑iAi†Ai=IN.On the other hand, if the Kraus operators ofΦ𝒜 satisfy the dual condition(2.6)∑iAiAi†=IN,
then Φ𝒜 is said to beunital. Note that (2.6) is equivalent to the simple statement that Φ𝒜(𝕀N)=𝕀N.A quantum operation which is both unital and trace preserving is calledbistochastic. The term doubly stochastic quantum channel also has been used to refer to quantum operations of this sort [23]. Note that a bistochastic quantum operation transforms elements of 𝔇(ℋ) into elements of 𝔇(ℋ). In other words, since the elements of 𝔇(ℋ) represent the states of a quantum system, a bistochastic quantum operation transforms states of a quantum system into other legitimate states of that system.


## 3. Limit Theorems for Bistochastic Quantum Operations

In the sequel, the proofs of all theorems, corollaries and supporting lemmas are deferred to Section4.By [24], a bistochastic quantum operation Φ𝒜 must satisfy the condition ∥Φ𝒜∥≤1. Thus the spectrum of Φ𝒜 is confined to the unit disk. Also, since Φ𝒜(𝕀N)=𝕀N, we see that λ=1 is an eigenvalue of Φ𝒜 and ∥Φ𝒜∥=1. For future reference, we record these observations in the form of a lemma.Lemma 3.1.
LetΦ𝒜 be a bistochastic quantum operation on the Hilbert space 𝔅(ℋ), then  (1)
∥Φ𝒜∥=1;(2)
ifλ is an eigenvalue of Φ𝒜, then |λ|≤1;(3)
the valueλ=1 is an eigenvalue of Φ𝒜.The observations recorded in Lemma3.1 are new by no means. See, for instance, [25].For an eigenvalueλ of Φ𝒜, let Ker(Φ𝒜-λ𝕀) and Ran(Φ𝒜-λ𝕀) denote, respectively, the kernel and range of the operator Φ𝒜-λ𝕀 on the Hilbert space 𝔅(ℋ).Lemma 3.2.
LetΦ𝒜 be a bistochastic quantum operation on 𝔅(ℋ), and let λ be an eigenvalue of Φ𝒜 with |λ|=1. Then Ker(Φ𝒜-λ𝕀)∩Ran(Φ𝒜-λ𝕀)={0}.From the preceding lemma, we can derive an important inference concerning the algebraic and geometric multiplicities of the eigenvalues ofΦ𝒜. For an eigenvalue λ of Φ𝒜, let its algebraic multiplicity be denoted by m(λ), and let its geometric multiplicity be denoted by g(λ). Recall that g(λ)=dimKer(Φ𝒜-λ𝕀). In addition, let the spectrum of Φ𝒜 be denoted by Λ(Φ𝒜), and let Λ1(Φ𝒜) denote the subset of Λ(Φ𝒜) consisting of λ∈Λ(Φ𝒜) with |λ|=1.Lemma 3.3.
Ifλ∈Λ1(Φ𝒜), where Φ𝒜 is a bistochastic quantum operation on 𝔅(ℋ), then m(λ)=g(λ).As in the proof of Lemma3.3, let [Φ𝒜] denote the N2×N2 Jordan canonical matrix representation of Φ𝒜. To conserve type-set space, and without undue risk of confusion to the reader, we prefer to display the matrix [Φ𝒜] in the following self-explanatory format: (3.1)[ΦA]=diag⁡(λ1,λ2,…,λk,J1,J2,…,Jh),
where λi∈Λ1(Φ𝒜) and Jr, r=1,2,…,h, denote the Jordan blocks corresponding to eigenvalues whose norms are strictly less than unity.For an eigenvalueλ of Φ𝒜, let EΦ𝒜(λ)=Ker(Φ𝒜-λ𝕀) denote the eigenspace of λ. In particular, for λ=1, it has been established in [26–28] that (3.2)EΦA(1)=Ker⁡(ΦA-I)={X∈B(H):XAi=AiX;i=1,2,…}.The following lemma articulates the special status ofEΦ𝒜(1) relative to the other eigenspaces of Φ𝒜.Lemma 3.4.
Letλ be an eigenvalue of Φ𝒜 such that |λ|=1 and λ≠1. Let α be an eigenvalue of Φ𝒜 with |α|<1. Let Y1,…,Yjα denote the generalized eigenvectors belonging to α. Then  (1)
EΦ𝒜(1)⊥EΦ𝒜(λ);(2)
EΦ𝒜(1)⊥Span{Y1,…,Yjα}.In other words, by Lemma3.4, the eigenspace EΦ𝒜(1) is orthogonal to every eigenvector (including every generalized eigenvector) of Φ𝒜 belonging to eigenvalues other than λ=1.Theorem 3.5.
LetΦ𝒜 be a bistochastic quantum operation on the Hilbert space 𝔅(ℋ), and let ρ(0)∈𝔇(ℋ) denote the density matrix representing the initial state of a quantum system. Let g(1) denote the geometric multiplicity of the eigenvalue λ=1, and let {Zr}1≤r≤g(1) denote an orthonormal basis for EΦ𝒜(1). If λ=1 is the only eigenvalue of Φ𝒜 on the unit circle, then the iterated succession of quantum states ρ(t)=Φ𝒜tρ(0) converges to ρ(∞)=∑l=1g(1)tr(Zl†ρ(0))Zl. In particular, if g(1)=1, then limt→∞Φ𝒜tρ(0)=(1/N)𝕀, independently of the initial state ρ(0).In previous publications, such as [29], the special case of a bistochastic quantum operation whereby g(1)=1 is known as a primitive quantum channel. Reference [29] provides an excellent analysis of this line of investigation.In the literature, numerous examples can be found of bistochastic quantum operations (a.k.a. quantum channels) for which the limiting behavior of the associated quantum Markov chain is governed by Theorem3.5. Among these examples are (1) the quantum walk on a cycle, subject to decoherence on the degree freedom of coin [30] and (2) the generalized depolarizing channel on a multiple-dimensional quantum system [19, 20]. These examples can be treated as special cases of the following corollary to Theorem 3.5.Corollary 3.6.
If the bistochastic quantum operationΦ𝒜 is given by
(3.3)ΦA(X)=(1-p)X+∑iAiXAi†,
where ∑iAiAi†=p𝕀 and 0<p<1, then, as in Theorem 3.5, ρ(∞)=∑r=1g(1)tr(Zr†ρ(0))Zr.In the context of quantum channels, the Kraus operators defining the generalizeddepolarizing channel can be scalar multiples of the discrete Weyl operators (or the generalized Pauli operators). When such is the case, as in, for instance, [23], the eigenspace of the eigenvalue 1 must be one-dimensional. Consequently the associated quantum Markov chain must converge to the maximally mixed state, regardless of the initial state.The main assertion of Theorem3.5 pertains only to bistochastic quantum operations Φ𝒜 possessing no eigenvalues on the unit circle other than λ=1. In general, when this condition is relaxed to admit other eigenvalues on the unit circle, the expression ρ(t)=Φ𝒜tρ(0) is compelled no longer to converge to a stationary state. Many examples can be found in the literature, including classic cases of unitary quantum walks on finite graphs [1].However, even if a limiting state fails to exist in the usual sense, we still might want to probe the possibility of a “limiting state”ρ̃(∞) in the sense of Cesàro: (3.4)ρ̃(∞)=lim⁡t→∞ΦAρ(0)+ΦA2ρ(0)+⋯+ΦAtρ(0)t.In terms of this generalized sense of “limiting state,” it turns out that every quantum Markov chain converges.Theorem 3.7.
LetΦ𝒜 be a bistochastic quantum operation on the Hilbert space 𝔅(ℋ), and let ρ(0)∈𝔇(ℋ) denote the density matrix representing the initial state of a quantum system. Let g(1) denote the geometric multiplicity of the eigenvalue λ=1, and let {Zr}1≤r≤g(1) denote an orthonormal basis for EΦ𝒜(1). Then, for every ρ(0)∈𝔇(ℋ),
(3.5)ρ̃(∞)=lim⁡t→∞1t∑n=1tΦAnρ(0)=∑l=1g(1)tr⁡(Zl†ρ(0))Zl.According to this theorem, the sequence of Cesàro means{(1/t)∑n=1tΦ𝒜n}t=1∞ is guaranteed always to converge to a limit which coincides with the orthogonal projection of the initial state upon the eigenspace of the unit eigenvalue of the bistochastic quantum operation. It is only fair to point out that alternate versions of this result have appeared in the literature. For instance, in  [26, Theorem  2.4(a)] it is shown that there exists a subsequence of the full sequence of Cesàro means which converges to a limit within the space of fixed points of the quantum operation. A closer match to Theorem 3.7, derived by different means and framed in different language, can be found in [31, Chapter 6].As an immediate corollary of Theorem3.7, it follows, as in [1], that the time-averaged probability distribution for any unitary quantum walk on a finite graph must converge. For a unitary quantum walk on a finite graph starting with an initial state |α0〉, let Pn(v∣α0) denote the probability of finding the walker at node v at time n, and let P̅t(v∣α0)=(1/t)∑n=1tPn(v∣α0) denote the time-averaged probability.Corollary 3.8 (see [1, Theorem  3.4]).
Letψj,λj denote the unit eigenvectors and corresponding eigenvalues of the unitary operator U associated with a unitary quantum walk on a finite graph, and let the bistochastic quantum operation Φ𝒜 be defined by Φ𝒜(ρ)=UρU†. Then, for any initial state |α0〉=∑jaj|ψj〉, one has
(3.6)lim⁡t→∞P̅t(v∣α0)=∑i,j,aaiaj⋆〈a,v∣ψi〉〈ψj∣a,v〉,
where the sum is restricted to pairs i,j such that λi=λj.


## 4. Proofs of the Theorems

This section is reserved for the proofs of theorems, lemmas, and corollaries given in Section3.Proof of Lemma3.2.
SupposeX∈Ker(Φ𝒜-λ𝕀)∩Ran(Φ𝒜-λ𝕀). Then Φ𝒜(X)=λX and Φ𝒜(Y)-λY=X for some Y∈𝔅(ℋ). Applying the linearity of Φ𝒜, we infer that Φ𝒜n(Y)=λnY+nλn-1X for all n≥1. Consequently ∥nλn-1X∥=∥Φ𝒜n(Y)-λnY∥, which implies that n∥X∥≤∥Φ𝒜∥n∥Y∥+∥Y∥. But since ∥Φ𝒜∥=1 (see Lemma 3.1), we have n∥X∥≤2∥Y∥. Since this inequality must hold for all n≥1, we conclude that ∥X∥=0, whence X=0.In the above proof, we have borrowed liberally from the reasoning employed in [18], which treats the special case of a random unitary operation.Proof of Lemma3.3.
We proceed by contradiction. Supposem(λ)≠g(λ). Let [Φ𝒜] denote the N2×N2 matrix representation of Φ𝒜 in Jordan canonical form. On the one hand, [Φ𝒜] must contain a Jordan block J belonging to λ of size >1. On the other hand, by standard matrix theory, there must exist a generalized eigenvector, say v, of Φ𝒜 such that (Φ𝒜-λ𝕀)v is itself an eigenvector of Φ𝒜. This implies that Ker(Φ𝒜-λ𝕀)∩Ran(Φ𝒜-λ𝕀)≠{0}, which contradicts Lemma 3.2.Proof of Lemma3.4.
The adjoint operator ofΦ𝒜 is given by
(4.1)ΦA†(X)=∑iAi†XAi.
Thus, by (3.2), we have Ker(Φ𝒜-𝕀)=Ker(Φ𝒜†-𝕀).
To prove (1) in Lemma3.4, let Z∈Ker(Φ𝒜-𝕀) and Y∈Ker(Φ𝒜-λ𝕀). Then 〈Z,Y〉=〈Φ𝒜†Z,Y〉=〈Z,Φ𝒜Y〉=λ〈Z,Y〉. Since λ≠1, it follows that 〈Z,Y〉=0. Hence Ker(Φ𝒜-𝕀)⊥Ker(Φ𝒜-λ𝕀).
We proceed to justify (2) in Lemma3.4. For an eigenvalue α with |α|<1, we may assume, without loss of generality, that the generalized eigenvectors belonging to α are arranged in a sequence Y1,Y2,…,Yjα such that
(4.2)(ΦA-αI)Yr=Yr-1,
where, by definition, Y0=0. It follows that (Φ𝒜-α𝕀)rYr=0. If Z∈Ker(Φ𝒜-𝕀), then 〈Z,Y1〉=〈Φ𝒜†Z,Y1〉=〈Z,Φ𝒜Y1〉=α〈Z,Y1〉, which implies that 〈Z,Y1〉=0. Similarly, 〈Z,Y2〉=〈Φ𝒜†Z,Y2〉=〈Z,Φ𝒜Y2〉=〈Z,Y1〉+α〈Z,Y2〉, from which it follows that 〈Z,Y2〉=0. Likewise, by the same reasoning, applied repeatedly, we deduce that 〈Z,Yr〉=0 for any r. Thus Ker(Φ𝒜-𝕀)⊥Span{Y1,Y2,…,Yjα}.Proof of Theorem3.5.
Letα1,…,αh denote the eigenvalues of Φ𝒜 whose absolute values are less than 1. Since, by Lemma 3.3, λ=1 is the only eigenvalue on the unit circle, a basis for 𝔅(ℋ) might be assembled as follows. Starting with an orthonormal basis for eigenspace EΦ𝒜(1), append a maximal set of linearly independent generalized eigenvectors belonging to the eigenvalues αr, r=1,2,…,h. Then, relative to this basis, the Jordan canonical matrix representation of Φ𝒜 is given by
(4.3)[ΦA]=diag⁡(Ig(1),J1,J2,…,Jh),
where Jr is the Jordan block corresponding to the eigenvalue αr, r=1,2,…,h.
Consider what becomes of the Jordan blocks of the powers[Φ𝒜]t as t→∞. Since each of the Jordan blocks Jr is an upper triangular matrix whose diagonal is populated by a single eigenvalue of modulus strictly less than unity, it is a simple exercise in elementary algebra to show that limt→∞Jrt=Or (zero matrix of same size as Jr). Thus, if we define
(4.4)[ΦA∞]=diag⁡(Ig(1),O1,O2,…,Oh),
then ∥Φ𝒜t-Φ𝒜∞∥→0.
Next, we consider the effect upon on the initial stateρ(0) of Φ𝒜t as t→∞.
By Lemma3.4, we may choose for 𝔅(ℋ) a basis consisting of a basis Z1,Z2,…,Zg(1) for EΦ𝒜(1) together with an orthogonally complement basis consisting of generalized eigenvectors belonging to all other eigenvalues. In terms of such a basis we have ρ(0)=∑r=1g(1)crZr⊕W, where W⊥Zr for all r=1,2,…,g(1). By simple linear algebra, it follows that cr=tr(Zr†ρ(0)) for r=1,2,…,g(1). Therefore, by (4.4), we have limt→∞Φ𝒜tρ(0)=∑r=1g(1)tr(Zr†ρ(0))Zr.
Finally, ifdimKer(Φ𝒜-𝕀)=1, then, to serve as a normalized basis for EΦ𝒜(1), we may take {(1/N)𝕀N}, from which it follows that limt→∞Φ𝒜tρ(0)=(1/N)𝕀, for any initial state ρ(0).Proof of Corollary3.6.
It suffices to verify thatλ=1 is the only eigenvalue of Φ𝒜 on the unit circle. By [24], whenever ∑iAiAi†=p𝕀, then ∥∑iAiXAi†∥≤p∥X∥ for any operator X. Now suppose λ is an eigenvalue of Φ𝒜 on the unit circle and X is an eigenvector belonging to λ. Then Φ𝒜(X)=λX=(1-p)X+∑iAiXAi†, from which it follows that ∥λX∥=∥X∥=∥(1-p)X+∑iAiXAi†∥≤∥(1-p)X∥+∥∑iAiXAi†∥≤(1-p)∥X∥+p∥X∥=∥X∥. Thus, all of the preceding inequalities actually are equalities. In particular, we have ∥∑iAiXAi†∥=p∥X∥. Thus (λ-1+p)X=∑iAiXAi†, which implies that |λ-1+p|·∥X∥=p∥X∥. It follows that λ=1.Proof of Theorem3.7.
To evaluate the limit in (3.4), we have only to reexamine (3.1). For every eigenvalue λ=λi of Φ𝒜, if λ≠1, then (1/t)∑n=1tλn=(1/t)(λ-λt+1/1-λ)→0. Similarly, for every Jordan block J=Jr in (3.1), we have (1/t)∑n=1tJn=(1/t)(𝕀-J)-1(J-Jt+1)→0. Based on these two observations, we deduce that (1/t)∑n=1t[Φ𝒜]n converges to the diagonal matrix
(4.5)Φ̅A∞=[Ig(1)000].
The expression for ρ̃(∞) follows by a path of reasoning identical to that employed in the proof of Theorem 3.5 to obtain the analogous expression for the limiting state in that situation.Proof of Corollary3.8.
As an orthonormal basis for the eigenspaceEΦ𝒜(1), we may take the set of vectors ⋃i,j{|ψi〉〈ψj|}, where the union is restricted to pairs i,j such that λi=λj. Accordingly, the orthogonal projection of the initial state ρ(0)=|α0〉〈α0| into the eigenspace EΦ𝒜(1) is
(4.6)∑i,j〈|ψi〉〈ψj|,ρ(0)〉|ψi〉〈ψj|=∑i,jαiαj⋆|ψi〉〈ψj|,
where, as above, the sum is restricted to pairs i,j such that λi=λj.
Since(4.7)P̅t(v∣α0)=∑atr⁡(|a,v〉〈a,v|1t∑n=1tΦAn|α0〉〈α0|)
and since ∑atr(|a,v〉〈a,v|·) is a continuous function of the argument, it follows, by Theorem 3.7, that P̅t(v∣α0) converges to
(4.8)∑atr⁡(|a,v〉〈a,v|∑i,jαiαj⋆|ψi〉〈ψj|)=∑i,j,aaiaj⋆〈a,v∣ψi〉〈ψj∣a,v〉,
where the sum is restricted to pairs i,j such that λi=λj.


## 5. Classification of Bistochastic Quantum Operations

LetΦ𝒜:𝔅(ℋ)→𝔅(ℋ) be a bistochastic quantum operation, and let ρ(0)∈𝔇(ℋ) denote the density matrix representing the initial state of a quantum system. Depending on the limiting behavior, as t→∞, of the corresponding quantum Markov process, Φ𝒜 must belong to one of the following four mutually exclusive categories:(1)
limt→∞Φ𝒜tρ(0) converges to the maximally mixed state (1/N)𝕀, independently of the initial state ρ(0);(2)
limt→∞Φ𝒜tρ(0) converges, but the limit depends upon the initial state ρ(0);(3)
limt→∞Φ𝒜tρ(0) fails to converge, but the Cesàro average limt→∞(1/t)∑n=1tΦ𝒜nρ(0) exists and equals the maximally mixed state (1/N)𝕀, independently of the initial state ρ(0);(4)
limt→∞Φ𝒜tρ(0) fails to converge, but the Cesàro average limt→∞(1/t)∑n=1tΦ𝒜nρ(0) exists and depends upon the initial state ρ(0).To elucidate each of the above categories, we proceed to offer some comments and examples.In category (1), the quantum operationΦ𝒜 possesses no eigenvalue on the unit circle other than λ=1. Moreover, the eigenspace Ker(Φ𝒜-𝕀) of λ=1 is one-dimensional, spanned only by the density operator (1/N)𝕀, or contains only a single density operator (1/N)𝕀.As an example, consider the quantum operation [16] defined by (5.1)ΦA(X)=12X+12UXU†,
where U is the unitary transformation given by (5.2)U=[12-323212].According to [18], this is an example of a so-calledrandom unitary operation. The eigenspace Ker(Φ𝒜-λ𝕀) where λ∈Λ1(Φ𝒜) is equal to the set Dλ:={X∈𝔅(ℋ):𝕀X=λX𝕀andUX=λXU}. Evidently, Λ1(Φ𝒜)={1}. A simple calculation shows that the eigenspace Ker(Φ𝒜-𝕀) of the eigenvalue λ=1 is D1={k𝕀:k∈ℂ}, which is one-dimensional. Therefore limt→∞Φ𝒜tX=(1/2)𝕀2 for any initial state X.It can be verified that thebit-phase flip channel [19, page 377] also belongs to category (1). A less trivial example of this sort can be found in [32].In category (2),Φ𝒜 possesses as its only eigenvalue on the unit circle the value λ=1 and Ker(Φ𝒜-𝕀) is at least two-dimensional.As an example, consider the quantum operationΦ𝒜 associated with a so-called phase flip channel on single qubits [19], given by (5.3)ΦA(X)=pX+(1-p)ZXZ†,
where Z is the Pauli matrix: (5.4)Z=[100-1].By a pattern of reasoning similar to that employed in the previous example, we infer thatΛ1(Φ𝒜)={1}, and the eigenspace Ker(Φ𝒜-𝕀) is spanned by the two linearly independent density operators: (5.5)X1=[1000],X2=[0001].In this case,limt→∞Φ𝒜tX=X∞, where X and X∞ are given by(5.6)X=[a11a12a21a22],X∞=[a1100a22].
It can be verified without much difficulty that the bit flip channel [19, page 376] belongs also to category (2). Additional examples of this sort can be found in [20, page 108].In category (3), the quantum operationΦ𝒜 possesses at least two distinct eigenvalues (including λ=1) on the unit circle. The eigenspace of λ=1, namely, Ker(Φ𝒜-𝕀), is one-dimensional and spanned by 𝕀. An example of this type of quantum operation is provided by quantum walks on the N-cycle. In this scenario, N is assumed even, the Hadamard transform serves as the coin operator, and the evolution of the system is subject to decoherence on both the position and the coin degrees freedom. A detailed treatment of this model is planned for our forthcoming paper [33].In category (4),Φ𝒜 has at least two distinct eigenvalues (including λ=1) on the unit circle and Ker(Φ𝒜-𝕀) contains at least two linearly independent density operators. As an example of this type of quantum operation, we cite [18] which studies the properties of the so-called two-qubit controlled-not operators.


## 6. Conclusion and Related Questions

We speculate that the eigenspaces of  eigenvalues on the unit circle, might conform always to a formulation in terms of a set of “Kraus operators”, and this formulation might provide an efficient means for identifying the eigenspaces of all eigenvalues of absolute 1. Forrandom unitary operations, as in [18], and so-called generalized random unitary operations,as in [33], the structures of the aforementioned eigenspaces have been fully elaborated.



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*Source: 740816-2011-08-25.xml*