Quantum Markov Chains: Unraveling Quantum Dynamics and Information Loss

Quantum Markov Chains (QMCs) are a recent trend in quantum information theory, used to describe open quantum dynamics on graphs. They can model the evolution of quantum systems and the loss of information to the environment. This article focuses on continuous-time QMCs, which allow for exact probability calculations. The authors, Manuel D De La Iglesia and Carlos F Lardizabal, explore the use of QMCs in a range of settings, particularly where Lindblad generators are induced by a single positive map. They also discuss the potential for future research in this area, particularly in relation to open quantum walks and QMCs.

What are Quantum Markov Chains and How Do They Work?

Quantum Markov Chains (QMCs) are positive maps on a trace-class space that describe open quantum dynamics on graphs. They share a statistical resemblance with classical random walks, but also allow for internal quantum degrees of freedom. This article focuses on continuous-time QMCs on the integer line, half-line, and finite segments, which enable exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures.

The methods used in this study are applicable to a wide range of settings, but the authors, Manuel D De La Iglesia and Carlos F Lardizabal, have chosen to restrict their focus to classes of examples where the Lindblad generators are induced by a single positive map. This is done so that the Stieltjes transforms of the measures and their inverses can be calculated explicitly.

The study of quantum versions of random walks on graphs is a recent trend in quantum information theory. This is achieved by taking a classical model as a starting point and investigating statistical notions that are applicable to quantum settings. The discrete-time coined quantum walk on the line is a popular model for unitary evolutions on the integer line, and the case of continuous-time walks provides valuable insight into the statistics of quantum evolutions described by 1-parameter unitary groups.

How Do Quantum Markov Chains Model Information Loss?

In addition to modeling the evolution of quantum systems, Quantum Markov Chains can also model the loss of information to the environment in both discrete and continuous time. This is particularly relevant in the context of open dissipative quantum versions of random walks.

The authors’ motivation for this study comes from a class of examples seen in the classical theory of Markov processes. A tridiagonal matrix, known as the generator or infinitesimal operator of a birth-death process on the half-line, can be used to calculate the probability of finding a particle at a specific vertex in terms of polynomials.

Favard’s Theorem guarantees that there exists at least one probability measure supported on the interval 0 to infinity such that the polynomials defined above are orthogonal with respect to it. The Karlin-McGregor formula can then be used to calculate the probability of being at a certain vertex at a given time, given that the process has started at another vertex.

What are the Applications of Quantum Markov Chains?

The authors are interested in open quantum versions of this classical setting. More precisely, they consider the context of Quantum Markov Chains as defined by S Gudder, where an operator specifies transitions between vertices of any fixed graph. Such transition effects are given by positive maps acting on appropriate spaces of density matrices.

In the 1-dimensional cases studied in this work, the Quantum Markov Chain can be described by block tridiagonal matrices, generalizing the structure of the generator of a birth-death process. The authors recall recent results on such matters, including the problem of finding matrix-valued measures for discrete-time Open Quantum Walks (OQWs) on the integer half-line and the consideration of matrix-valued measures for discrete-time Quantum Markov Chains on the integer line, half-line, and finite segments.

How Does the Underlying Classical Structure Influence Quantum Markov Chains?

If the transition operators of a Quantum Markov Chain are relatively simple, such as corresponding to Hermitian matrices, it is possible to obtain quantum information of the system in terms of some underlying classical polynomial structure. Under certain conditions, a linear change of coordinates on the components of the Quantum Markov Chain will allow this question to be answered positively and to obtain non-classical information on the walk.

In practical terms, with a quantum version of the Karlin-McGregor formula, one can obtain transition probabilities for Quantum Markov Chains. In previous works, such a formula is deduced and explored to some extent, but in practice, the calculations mostly concerned the case of site recurrence. In this work, the authors discuss a proper setting of Quantum Markov Chains where they are able to present closed formulae for arbitrary transition probabilities.

What are the Future Directions for Research on Quantum Markov Chains?

The authors focus on the simplest non-classical instances of Quantum Markov Chains, namely Hermitian walks of one qubit moving on the integer line, half-line, finite segment, and present a brief systematic discussion on such settings. They consider translation-invariant walks, meaning that they have the same transition rule on every vertex on the half-line and the finite segment, specifying the proper boundary conditions.

This kind of description has not been explored in the context of Open Quantum Walks and Quantum Markov Chains so far, and the authors believe it opens up new avenues for future research. The methods and findings presented in this work could be applied to a wide range of settings, providing valuable insights into the dynamics and statistics of quantum systems.