On Limiting Distributions of Quantum Markov Chains

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).

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-called quantum 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 a quantum 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, if 𝜆=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 (Section 2), 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 space ℋ of finite dimension 𝑁, let 𝔅(ℋ) denote the set of all linear operators on ℋ, with inner product defined by ⟨𝑋,𝑌⟩≡tr𝑋^†𝑌. (2.1)

The corresponding norm, called Frobenius norm or Schatten 2- norm, is defined by ‖𝑋‖≡tr𝑋^†𝑋^1/2=⟨𝑋,𝑋⟩. (2.2) This choice of norm on 𝔅(ℋ) will remain in effect throughout this paper.

Let 𝔇(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 a superoperator Φ on 𝔅(ℋ), we mean a linear mapping Φ∶𝔅(ℋ)→𝔅(ℋ), with norm defined by ‖𝚽‖≡Sup_{𝑋∈𝔅(ℋ)}‖𝚽(𝑋)‖‖𝑋‖. (2.3)

Note that dim𝔅(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 superoperator Φ must be completely positive, meaning that the extended map Φ⊗𝕀_𝑛 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={𝐴_𝑖∣𝑖=1,2,…,𝑁²} of “Kraus operators” as follows:𝚽_𝒜(𝑋)=_𝑖𝐴_𝑖𝑋𝐴^†_𝑖. (2.4) In this expression, which we call the “Choi expansion” of Φ, the symbol 𝐴^†_𝑖 denotes 𝐴^𝑇_𝑖 (transpose of the complex conjugate of 𝐴_𝑖).

In terms of the Choi expansion, the condition of being trace preserving, meaning that Tr(Φ_𝒜(𝑋))=Tr(𝑋) for all 𝑋∈𝔅(ℋ), is equivalent to the condition_𝑖𝐴^†_𝑖𝐴_𝑖=𝕀_𝑁. (2.5)

On the other hand, if the Kraus operators of Φ_𝒜 satisfy the dual condition_𝑖𝐴_𝑖𝐴^†_𝑖=𝕀_𝑁, (2.6) then Φ_𝒜 is said to be unital. Note that (2.6) is equivalent to the simple statement that Φ_𝒜(𝕀_𝑁)=𝕀_𝑁.

A quantum operation which is both unital and trace preserving is called bistochastic. 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 Section 4.

By [24], a bistochastic quantum operation Φ_𝒜 must satisfy the condition ‖Φ_𝒜‖≤1. Thus the spectrum of Φ_𝒜 is confined to the unit disk. Also, since Φ_𝒜(𝕀_𝑁)=𝕀_𝑁, we see that 𝜆=1 is an eigenvalue of Φ_𝒜 and ‖Φ_𝒜‖=1. For future reference, we record these observations in the form of a lemma.

Lemma 3.1. Let Φ_𝒜 be a bistochastic quantum operation on the Hilbert space 𝔅(ℋ), then (1)‖Φ_𝒜‖=1; (2)if 𝜆 is an eigenvalue of Φ_𝒜, then |𝜆|≤1; (3)the value 𝜆=1 is an eigenvalue of Φ_𝒜.

The observations recorded in Lemma 3.1 are new by no means. See, for instance, [25].

For an eigenvalue 𝜆 of Φ_𝒜, let Ker(Φ_𝒜−𝜆𝕀) and Ran(Φ_𝒜−𝜆𝕀) denote, respectively, the kernel and range of the operator Φ_𝒜−𝜆𝕀 on the Hilbert space 𝔅(ℋ).

Lemma 3.2. Let Φ_𝒜 be a bistochastic quantum operation on 𝔅(ℋ), and let 𝜆 be an eigenvalue of Φ_𝒜 with |𝜆|=1. Then Ker(Φ_𝒜−𝜆𝕀)∩Ran(Φ_𝒜−𝜆𝕀)={0}.

From the preceding lemma, we can derive an important inference concerning the algebraic and geometric multiplicities of the eigenvalues of Φ_𝒜. For an eigenvalue 𝜆 of Φ_𝒜, let its algebraic multiplicity be denoted by 𝑚(𝜆), and let its geometric multiplicity be denoted by 𝑔(𝜆). Recall that 𝑔(𝜆)=dimKer(Φ_𝒜−𝜆𝕀). In addition, let the spectrum of Φ_𝒜 be denoted by Λ(Φ_𝒜), and let Λ₁(Φ_𝒜) denote the subset of Λ(Φ_𝒜) consisting of 𝜆∈Λ(Φ_𝒜) with |𝜆|=1.

Lemma 3.3. If 𝜆∈Λ₁(Φ_𝒜), where Φ_𝒜 is a bistochastic quantum operation on 𝔅(ℋ), then 𝑚(𝜆)=𝑔(𝜆).

As in the proof of Lemma 3.3, let [Φ_𝒜] denote the 𝑁²×𝑁² Jordan canonical matrix representation of Φ_𝒜. To conserve type-set space, and without undue risk of confusion to the reader, we prefer to display the matrix [Φ_𝒜] in the following self-explanatory format: 𝚽_𝒜=diag𝜆₁,𝜆₂,…,𝜆_𝑘,𝐽₁,𝐽₂,…,𝐽_ℎ, (3.1) where 𝜆_𝑖∈Λ₁(Φ_𝒜) and 𝐽_𝑟, 𝑟=1,2,…,ℎ, denote the Jordan blocks corresponding to eigenvalues whose norms are strictly less than unity.

For an eigenvalue 𝜆 of Φ_𝒜, let 𝐸_Φ𝒜(𝜆)=Ker(Φ_𝒜−𝜆𝕀) denote the eigenspace of 𝜆. In particular, for 𝜆=1, it has been established in [26–28] that 𝐸_𝚽𝒜(1)=Ker𝚽_𝒜−𝕀=𝑋∈𝔅(ℋ)∶𝑋𝐴_𝑖=𝐴_𝑖𝑋;𝑖=1,2,…. (3.2)

The following lemma articulates the special status of 𝐸_Φ𝒜(1) relative to the other eigenspaces of Φ_𝒜.

Lemma 3.4. Let 𝜆 be an eigenvalue of Φ_𝒜 such that |𝜆|=1 and 𝜆≠1. Let 𝛼 be an eigenvalue of Φ_𝒜 with |𝛼|<1. Let 𝑌₁,…,𝑌_𝑗𝛼 denote the generalized eigenvectors belonging to 𝛼. Then (1)𝐸_Φ𝒜(1)⟂𝐸_Φ𝒜(𝜆); (2)𝐸_Φ𝒜(1)⟂Span{𝑌₁,…,𝑌_𝑗𝛼}.

In other words, by Lemma 3.4, the eigenspace 𝐸_Φ𝒜(1) is orthogonal to every eigenvector (including every generalized eigenvector) of Φ_𝒜 belonging to eigenvalues other than 𝜆=1.

Theorem 3.5. Let Φ_𝒜 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 {𝑍_𝑟}_{1≤𝑟≤𝑔(1)} denote an orthonormal basis for 𝐸_Φ𝒜(1). If 𝜆=1 is the only eigenvalue of Φ_𝒜 on the unit circle, then the iterated succession of quantum states 𝜌(𝑡)=Φ^𝑡_𝒜𝜌(0) converges to 𝜌(∞)=∑^𝑔(1)_𝑙=1tr(𝑍^†_𝑙𝜌(0))𝑍_𝑙. In particular, if 𝑔(1)=1, then lim_𝑡→∞Φ^𝑡_𝒜𝜌(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 Theorem 3.5. 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 operation Φ_𝒜 is given by 𝚽_𝒜(𝑋)=1−𝑝𝑋+_𝑖𝐴_𝑖𝑋𝐴^†_𝑖, (3.3) where ∑_𝑖𝐴_𝑖𝐴^†_𝑖=𝑝𝕀 and 0<𝑝<1, then, as in Theorem 3.5, 𝜌(∞)=∑^𝑔(1)_𝑟=1tr(𝑍^†_𝑟𝜌(0))𝑍_𝑟.

In the context of quantum channels, the Kraus operators defining the generalized depolarizing 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 Theorem 3.5 pertains only to bistochastic quantum operations Φ_𝒜 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)=Φ^𝑡_𝒜𝜌(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” 𝜌(∞) in the sense of Cesàro: 𝜌(∞)=lim_𝑡→∞𝚽_𝒜𝜌(0)+𝚽²_𝒜𝜌(0)+⋯+𝚽^𝑡_𝒜𝜌(0)𝑡. (3.4)

In terms of this generalized sense of “limiting state,” it turns out that every quantum Markov chain converges.

Theorem 3.7. Let Φ_𝒜 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 {𝑍_𝑟}_{1≤𝑟≤𝑔(1)} denote an orthonormal basis for 𝐸_Φ𝒜(1). Then, for every 𝜌(0)∈𝔇(ℋ), 𝜌(∞)=lim_𝑡→∞1𝑡^𝑡_𝑛=1𝚽^𝑛_𝒜𝜌(0)=^𝑔(1)_𝑙=1tr𝑍^†_𝑙𝜌(0)𝑍_𝑙. (3.5)

According to this theorem, the sequence of Cesàro means {(1/𝑡)∑^𝑡_𝑛=1Φ^𝑛_𝒜}^∞_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 Theorem 3.7, 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 𝑃_𝑛(𝑣∣𝛼₀) denote the probability of finding the walker at node 𝑣 at time 𝑛, and let 𝑃_𝑡(𝑣∣𝛼₀)=(1/𝑡)∑^𝑡_𝑛=1𝑃_𝑛(𝑣∣𝛼₀) denote the time-averaged probability.

Corollary 3.8 (see [1, Theorem  3.4]). Let 𝜓_𝑗,𝜆_𝑗 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 Φ_𝒜 be defined by Φ_𝒜(𝜌)=𝑈𝜌𝑈^†. Then, for any initial state |𝛼₀⟩=∑_𝑗𝑎_𝑗|𝜓_𝑗⟩, one has lim_𝑡→∞𝑃_𝑡𝑣∣𝛼₀=_{𝑖,𝑗,𝑎}𝑎_𝑖𝑎^⋆_𝑗𝑎,𝑣∣𝜓_𝑖𝜓_𝑗∣𝑎,𝑣, (3.6) where the sum is restricted to pairs 𝑖,𝑗 such that 𝜆_𝑖=𝜆_𝑗.

4. Proofs of the Theorems

This section is reserved for the proofs of theorems, lemmas, and corollaries given in Section 3.

Proof of Lemma 3.2. Suppose 𝑋∈Ker(Φ_𝒜−𝜆𝕀)∩Ran(Φ_𝒜−𝜆𝕀). Then Φ_𝒜(𝑋)=𝜆𝑋 and Φ_𝒜(𝑌)−𝜆𝑌=𝑋 for some 𝑌∈𝔅(ℋ). Applying the linearity of Φ_𝒜, we infer that Φ^𝑛_𝒜(𝑌)=𝜆^𝑛𝑌+𝑛𝜆^𝑛−1𝑋 for all 𝑛≥1. Consequently ‖𝑛𝜆^𝑛−1𝑋‖=‖Φ^𝑛_𝒜(𝑌)−𝜆^𝑛𝑌‖, which implies that 𝑛‖𝑋‖≤‖Φ_𝒜‖^𝑛‖𝑌‖+‖𝑌‖. But since ‖Φ_𝒜‖=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 Lemma 3.3. We proceed by contradiction. Suppose 𝑚(𝜆)≠𝑔(𝜆). Let [Φ_𝒜] denote the 𝑁²×𝑁² matrix representation of Φ_𝒜 in Jordan canonical form. On the one hand, [Φ_𝒜] 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 Φ_𝒜 such that (Φ_𝒜−𝜆𝕀)𝑣 is itself an eigenvector of Φ_𝒜. This implies that Ker(Φ_𝒜−𝜆𝕀)∩Ran(Φ_𝒜−𝜆𝕀)≠{0}, which contradicts Lemma 3.2.

Proof of Lemma 3.4. The adjoint operator of Φ_𝒜 is given by 𝚽^†_𝒜(𝑋)=_𝑖𝐴^†_𝑖𝑋𝐴_𝑖. (4.1) Thus, by (3.2), we have Ker(Φ_𝒜−𝕀)=Ker(Φ^†_𝒜−𝕀).
To prove (1) in Lemma 3.4, let 𝑍∈Ker(Φ_𝒜−𝕀) and 𝑌∈Ker(Φ_𝒜−𝜆𝕀). Then ⟨𝑍,𝑌⟩=⟨Φ^†_𝒜𝑍,𝑌⟩=⟨𝑍,Φ_𝒜𝑌⟩=𝜆⟨𝑍,𝑌⟩. Since 𝜆≠1, it follows that ⟨𝑍,𝑌⟩=0. Hence Ker(Φ_𝒜−𝕀)⟂Ker(Φ_𝒜−𝜆𝕀).
We proceed to justify (2) in Lemma 3.4. 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,…,𝑌_𝑗𝛼 such that 𝚽_𝒜−𝛼𝕀𝑌_𝑟=𝑌_𝑟−1, (4.2) where, by definition, 𝑌₀=0. It follows that (Φ_𝒜−𝛼𝕀)^𝑟𝑌_𝑟=0. If 𝑍∈Ker(Φ_𝒜−𝕀), then ⟨𝑍,𝑌₁⟩=⟨Φ^†_𝒜𝑍,𝑌₁⟩=⟨𝑍,Φ_𝒜𝑌₁⟩=𝛼⟨𝑍,𝑌₁⟩, which implies that ⟨𝑍,𝑌₁⟩=0. Similarly, ⟨𝑍,𝑌₂⟩=⟨Φ^†_𝒜𝑍,𝑌₂⟩=⟨𝑍,Φ_𝒜𝑌₂⟩=⟨𝑍,𝑌₁⟩+𝛼⟨𝑍,𝑌₂⟩, from which it follows that ⟨𝑍,𝑌₂⟩=0. Likewise, by the same reasoning, applied repeatedly, we deduce that ⟨𝑍,𝑌_𝑟⟩=0 for any 𝑟. Thus Ker(Φ_𝒜−𝕀)⟂Span{𝑌₁,𝑌₂,…,𝑌_𝑗𝛼}.

Proof of Theorem 3.5. Let 𝛼₁,…,𝛼_ℎ denote the eigenvalues of Φ_𝒜 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 𝐸_Φ𝒜(1), append a maximal set of linearly independent generalized eigenvectors belonging to the eigenvalues 𝛼_𝑟, 𝑟=1,2,…,ℎ. Then, relative to this basis, the Jordan canonical matrix representation of Φ_𝒜 is given by 𝚽_𝒜=diag𝕀_𝑔(1),𝐽₁,𝐽₂,…,𝐽_ℎ, (4.3) where 𝐽_𝑟 is the Jordan block corresponding to the eigenvalue 𝛼_𝑟, 𝑟=1,2,…,ℎ.
Consider what becomes of the Jordan blocks of the powers [Φ_𝒜]^𝑡 as 𝑡→∞. Since each of the Jordan blocks 𝐽_𝑟 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 lim_𝑡→∞𝐽^𝑡_𝑟=𝑂_𝑟 (zero matrix of same size as 𝐽_𝑟). Thus, if we define 𝚽^∞_𝒜=diag𝕀_𝑔(1),𝑂₁,𝑂₂,…,𝑂_ℎ, (4.4) then ‖Φ^𝑡_𝒜−Φ^∞_𝒜‖→0.
Next, we consider the effect upon on the initial state 𝜌(0) of Φ^𝑡_𝒜 as 𝑡→∞.
By Lemma 3.4, we may choose for 𝔅(ℋ) a basis consisting of a basis 𝑍₁,𝑍₂,…,𝑍_𝑔(1) for 𝐸_Φ𝒜(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)=∑^𝑔(1)_𝑟=1𝑐_𝑟𝑍_𝑟⊕𝑊, where 𝑊⟂𝑍_𝑟 for all 𝑟=1,2,…,𝑔(1). By simple linear algebra, it follows that 𝑐_𝑟=tr(𝑍^†_𝑟𝜌(0)) for 𝑟=1,2,…,𝑔(1). Therefore, by (4.4), we have lim_𝑡→∞Φ^𝑡_𝒜𝜌(0)=∑^𝑔(1)_𝑟=1tr(𝑍^†_𝑟𝜌(0))𝑍_𝑟.
Finally, if dimKer(Φ_𝒜−𝕀)=1, then, to serve as a normalized basis for 𝐸_Φ𝒜(1), we may take {(1/√𝑁)𝕀_𝑁}, from which it follows that lim_𝑡→∞Φ^𝑡_𝒜𝜌(0)=(1/𝑁)𝕀, for any initial state 𝜌(0).

Proof of Corollary 3.6. It suffices to verify that 𝜆=1 is the only eigenvalue of Φ_𝒜 on the unit circle. By [24], whenever ∑_𝑖𝐴_𝑖𝐴^†_𝑖=𝑝𝕀, then ‖∑_𝑖𝐴_𝑖𝑋𝐴^†_𝑖‖≤𝑝‖𝑋‖ for any operator 𝑋. Now suppose 𝜆 is an eigenvalue of Φ_𝒜 on the unit circle and 𝑋 is an eigenvector belonging to 𝜆. Then Φ_𝒜(𝑋)=𝜆𝑋=(1−𝑝)𝑋+∑_𝑖𝐴_𝑖𝑋𝐴^†_𝑖, from which it follows that ‖𝜆𝑋‖=‖𝑋‖=‖(1−𝑝)𝑋+∑_𝑖𝐴_𝑖𝑋𝐴^†_𝑖‖≤‖(1−𝑝)𝑋‖+‖∑_𝑖𝐴_𝑖𝑋𝐴^†_𝑖‖≤(1−𝑝)‖𝑋‖+𝑝‖𝑋‖=‖𝑋‖. Thus, all of the preceding inequalities actually are equalities. In particular, we have ‖∑_𝑖𝐴_𝑖𝑋𝐴^†_𝑖‖=𝑝‖𝑋‖. Thus (𝜆−1+𝑝)𝑋=∑_𝑖𝐴_𝑖𝑋𝐴^†_𝑖, which implies that |𝜆−1+𝑝|⋅‖𝑋‖=𝑝‖𝑋‖. It follows that 𝜆=1.

Proof of Theorem 3.7. To evaluate the limit in (3.4), we have only to reexamine (3.1). For every eigenvalue 𝜆=𝜆_𝑖 of Φ_𝒜, if 𝜆≠1, then (1/𝑡)∑^𝑡_𝑛=1𝜆^𝑛=(1/𝑡)(𝜆−𝜆^𝑡+1/1−𝜆)→0. Similarly, for every Jordan block 𝐽=𝐽_𝑟 in (3.1), we have (1/𝑡)∑^𝑡_𝑛=1𝐽^𝑛=(1/𝑡)(𝕀−𝐽)⁻¹(𝐽−𝐽^𝑡+1)→0. Based on these two observations, we deduce that (1/𝑡)∑^𝑡_𝑛=1[Φ_𝒜]^𝑛 converges to the diagonal matrix 𝚽^∞_𝒜=𝕀_𝑔(1)000. (4.5) The expression for 𝜌(∞) 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 Corollary 3.8. As an orthonormal basis for the eigenspace 𝐸_Φ𝒜(1), we may take the set of vectors ⋃_𝑖,𝑗{|𝜓_𝑖⟩⟨𝜓_𝑗|}, where the union is restricted to pairs 𝑖,𝑗 such that 𝜆_𝑖=𝜆_𝑗. Accordingly, the orthogonal projection of the initial state 𝜌(0)=|𝛼₀⟩⟨𝛼₀| into the eigenspace 𝐸_Φ𝒜(1) is _𝑖,𝑗𝜓_𝑖𝜓_𝑗,𝜌(0)𝜓_𝑖𝜓_𝑗=_𝑖,𝑗𝛼_𝑖𝛼^⋆_𝑗𝜓_𝑖𝜓_𝑗, (4.6) where, as above, the sum is restricted to pairs 𝑖,𝑗 such that 𝜆_𝑖=𝜆_𝑗.
Since 𝑃_𝑡𝑣∣𝛼₀=_𝑎||1tr𝑎,𝑣⟩⟨𝑎,𝑣𝑡^𝑡_𝑛=1𝚽^𝑛_𝒜𝛼₀𝛼₀ (4.7) and since ∑_𝑎tr(|𝑎,𝑣⟩⟨𝑎,𝑣|⋅) is a continuous function of the argument, it follows, by Theorem 3.7, that 𝑃_𝑡(𝑣∣𝛼₀) converges to tr|_𝑎𝑎,𝑣⟩⟨𝑎,𝑣|_𝑖,𝑗𝛼_𝑖𝛼^⋆_𝑗𝜓_𝑖𝜓_𝑗=_{𝑖,𝑗,𝑎}𝑎_𝑖𝑎^⋆_𝑗𝑎,𝑣∣𝜓_𝑖𝜓_𝑗∣𝑎,𝑣, (4.8) where the sum is restricted to pairs 𝑖,𝑗 such that 𝜆_𝑖=𝜆_𝑗.

5. Classification of Bistochastic Quantum Operations

Let Φ_𝒜∶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, Φ_𝒜 must belong to one of the following four mutually exclusive categories:(1)lim_𝑡→∞Φ^𝑡_𝒜𝜌(0) converges to the maximally mixed state (1/𝑁)𝕀, independently of the initial state 𝜌(0);(2)lim_𝑡→∞Φ^𝑡_𝒜𝜌(0) converges, but the limit depends upon the initial state 𝜌(0);(3)lim_𝑡→∞Φ^𝑡_𝒜𝜌(0) fails to converge, but the Cesàro average lim_𝑡→∞(1/𝑡)∑^𝑡_𝑛=1Φ^𝑛_𝒜𝜌(0) exists and equals the maximally mixed state (1/𝑁)𝕀, independently of the initial state 𝜌(0);(4)lim_𝑡→∞Φ^𝑡_𝒜𝜌(0) fails to converge, but the Cesàro average lim_𝑡→∞(1/𝑡)∑^𝑡_𝑛=1Φ^𝑛_𝒜𝜌(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 operation Φ_𝒜 possesses no eigenvalue on the unit circle other than 𝜆=1. Moreover, the eigenspace Ker(Φ_𝒜−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 𝚽_𝒜(𝑋)=121𝑋+2𝑈𝑋𝑈^†, (5.1) where 𝑈 is the unitary transformation given by 1𝑈=2−√32√3212. (5.2)

According to [18], this is an example of a so-called random unitary operation. The eigenspace Ker(Φ_𝒜−𝜆𝕀) where 𝜆∈Λ₁(Φ_𝒜) is equal to the set 𝐷_𝜆∶={𝑋∈𝔅(ℋ)∶𝕀𝑋=𝜆𝑋𝕀and𝑈𝑋=𝜆𝑋𝑈}. Evidently, Λ₁(Φ_𝒜)={1}. A simple calculation shows that the eigenspace Ker(Φ_𝒜−𝕀) of the eigenvalue 𝜆=1 is 𝐷₁={𝑘𝕀∶𝑘∈ℂ}, which is one-dimensional. Therefore lim_𝑡→∞Φ^𝑡_𝒜𝑋=(1/2)𝕀₂ for any initial state 𝑋.

It can be verified that the bit-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), Φ_𝒜 possesses as its only eigenvalue on the unit circle the value 𝜆=1 and Ker(Φ_𝒜−𝕀) is at least two-dimensional.

As an example, consider the quantum operation Φ_𝒜 associated with a so-called phase flip channel on single qubits [19], given by 𝚽_𝒜(𝑋)=𝑝𝑋+1−𝑝𝑍𝑋𝑍^†, (5.3) where 𝑍 is the Pauli matrix: 𝑍=100−1. (5.4)

By a pattern of reasoning similar to that employed in the previous example, we infer that Λ₁(Φ_𝒜)={1}, and the eigenspace Ker(Φ_𝒜−𝕀) is spanned by the two linearly independent density operators: 𝑋₁=1000,𝑋₂=0001. (5.5)

In this case, lim_𝑡→∞Φ^𝑡_𝒜𝑋=𝑋_∞, where 𝑋 and 𝑋_∞ are given by𝑎𝑋=¹¹𝑎₁₂𝑎₂₁𝑎₂₂,𝑋_∞=𝑎₁₁00𝑎₂₂. (5.6) 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 operation Φ_𝒜 possesses at least two distinct eigenvalues (including 𝜆=1) on the unit circle. The eigenspace of 𝜆=1, namely, Ker(Φ_𝒜−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), Φ_𝒜 has at least two distinct eigenvalues (including 𝜆=1) on the unit circle and Ker(Φ_𝒜−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. For random 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.

Acknowledgment

C. Liu was partially supported by NSF Grant CCF-1005564.