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).

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

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

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 [

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

Our results may represent substantial progress toward answering the first of a set of “open questions” posed by Ambainis [

In what follows, after a brief review of some preliminaries (Section

Given a Hilbert space

The corresponding norm, called

Let

By a

Note that

Among the set of superoperators, we distinguish a special subset called “quantum operations”. By definition, to qualify as a quantum operation, the superoperator

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 [

By Choi's Theorem [

In terms of the Choi expansion, the condition of being

On the other hand, if the Kraus operators of

A quantum operation which is both unital and trace preserving is called

In the sequel, the proofs of all theorems, corollaries and supporting lemmas are deferred to Section

By [

Let

if

the value

The observations recorded in Lemma

For an eigenvalue

Let

From the preceding lemma, we can derive an important inference concerning the algebraic and geometric multiplicities of the eigenvalues of

If

As in the proof of Lemma

For an eigenvalue

The following lemma articulates the special status of

Let

In other words, by Lemma

Let

In previous publications, such as [

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

If the bistochastic quantum operation

In the context of quantum channels, the Kraus operators defining the generalized

The main assertion of Theorem

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”

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

Let

According to this theorem, the sequence of Cesàro means

As an immediate corollary of Theorem

Let

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

Suppose

In the above proof, we have borrowed liberally from the reasoning employed in [

We proceed by contradiction. Suppose

The adjoint operator of

To prove (1) in Lemma

We proceed to justify (2) in Lemma

Let

Consider what becomes of the Jordan blocks of the powers

Next, we consider the effect upon on the initial state

By Lemma

Finally, if

It suffices to verify that

To evaluate the limit in (

As an orthonormal basis for the eigenspace

Since

Let

To elucidate each of the above categories, we proceed to offer some comments and examples.

In category (1), the quantum operation

As an example, consider the quantum operation [

According to [

It can be verified that the

In category (2),

As an example, consider the quantum operation

By a pattern of reasoning similar to that employed in the previous example, we infer that

In this case,

In category (3), the quantum operation

In category (4),

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

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