Quantum Markov Chains Research Articles

Analysis of the conditional mutual information in ballistic and diffusive non-equilibrium steady-states

The conditional mutual information (CMI) quantifies the amount of correlations shared between A and C given B. It therefore functions as a more general quantifier of bipartite correlations in multipartite scenarios, playing an important role in the theory of quantum Markov chains. In this paper we carry out a detailed study on the behavior of the CMI in non-equilibrium steady-states (NESS) of a quantum chain placed between two baths at different temperatures. These results are used to shed light on the mechanisms behind ballistic and diffusive transport regimes and how they affect correlations between different parts of a chain. We carry our study for the specific case of a 1D bosonic chain subject to local Lindblad dissipators at the boundaries. In addition, the chain is also subject to self-consistent reservoirs at each site, which are used to tune the transport between ballistic and diffusive. As a result, we find that the CMI is independent of the chain size L in the ballistic regime, but decays algebraically with L in the diffusive case. Finally, we also show how this scaling can be used to discuss the notion of local thermalization in non-equilibrium steady-states.

Quantum Markov chains: A unification approach

In this paper, we study a unified approach for quantum Markov chains (QMCs). A new quantum Markov property that generalizes the old one, is discussed. We introduce Markov states and chains on general local algebras, possessing a generic algebraic property. We stress that this kind of algebras includes both Boson and Fermi algebras. Our main results concern two reconstruction theorems for quantum Markov chains and for quantum Markov states. Namely, we illustrate the results through examples.

Recoverability from direct quantum correlations

We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons, we restrict our analysis to tree-structured sets of bipartite marginals. We focus on the tripartite scenario, where we solve the problem for the couples of measured marginals which are compatible with a quantum Markov chain, providing then an algebraic necessary and sufficient condition for the compatibility to be verified. We introduce the generalization of the procedure to the n-partite scenario, giving some preliminary results. In particular, we prove that if the pairwise Markov condition holds between the subparts then the choice of the best set of tree-structured bipartite marginals can be performed efficiently. Moreover, we provide a new characterization of quantum Markov chains in terms of quantum Bayesian updating processes.

Attractors and asymptotic dynamics of open discrete-time quantum walks on cycles

Open quantum walks often lead to a classical asymptotic behavior. Here, we look for a simple open quantum walk whose asymptotic behavior can be non-classical. We consider a discrete-time quantum walk on n-cycle subject to a random coin-dependent phase shift at a single position. This finite system, whose evolution is described by only two Kraus operators, can exhibit all kinds of asymptotic behavior observable in quantum Markov chains: it either evolves towards a maximally mixed state, or partially mixed state, or tends to an oscillatory motion on an asymptotic orbit. We find that the asymptotic orbits do not have a product structure, therefore the corresponding states can manifest entanglement between the position and the coin degrees of freedom, even if the system started in a product state.

Quantum Markov Chains: Recurrence, Schur Functions and Splitting Rules

In this work we study the recurrence problem for quantum Markov chains, which are quantum versions of classical Markov chains introduced by S. Gudder and described in terms of completely positive maps. A notion of monitored recurrence for quantum Markov chains is examined in association with Schur functions, which codify information on the first return to some given state or subspace. Such objects possess important factorization and decomposition properties which allow us to obtain probabilistic results based solely on those parts of the graph where the dynamics takes place, the so-called splitting rules. These rules also yield an alternative to the folding trick to transform a doubly infinite system into a semi-infinite one which doubles the number of internal degrees of freedom. The generalization of Schur functions --so-called FR-functions-- to the general context of closed operators in Banach spaces is the key for the present applications to open quantum systems. An important class of examples included in this setting are the open quantum random walks, as described by S. Attal et al., but we will state results in terms of general completely positive trace preserving maps. We also take the opportunity to discuss basic results on recurrence of finite dimensional iterated quantum channels and quantum versions of Kac's Lemma, in close association with recent results on the subject.

Factors Generated by XY-Model with Competing Ising Interactions on the Cayley Tree

In the present paper, we consider a quantum Markov chain corresponding to the XY-model with competing Ising interactions on the Cayley tree of order two. Earlier, it was proved that this state does exist and is unique. Moreover, it has clustering property. This means that the von Neumann algebra generated by this state is a factor. In the present paper, we establish that the factor generated by this state may have type \(\mathop {\mathrm{III_{\lambda }}}\)\(\lambda \in (0,1)\) which is unusual for states associated with models with nontrivial interactions.

Eigenbasis of the evolution operator of 2-tessellable quantum walks

Staggered quantum walks on graphs are based on the concept of graph tessellation and generalize some well-known discrete-time quantum walk models. In this work, we address the class of 2-tessellable quantum walks with the goal of obtaining an eigenbasis of the evolution operator. By interpreting the evolution operator as a quantum Markov chain on an underlying multigraph, we define the concept of quantum detailed balance, which helps to obtain the eigenbasis. A subset of the eigenvectors is obtained from the eigenvectors of the double discriminant matrix of the quantum Markov chain. To obtain the remaining eigenvectors, we have to use the quantum detailed balance conditions. If the quantum Markov chain has a quantum detailed balance, there is an eigenvector for each fundamental cycle of the underlying multigraph. If the quantum Markov chain does not have a quantum detailed balance, we have to use two fundamental cycles linked by a path in order to find the remaining eigenvectors. We exemplify the process of obtaining the eigenbasis of the evolution operator using the kagome lattice (the line graph of the hexagonal lattice), which has symmetry properties that help in the calculation process.

Open quantum random walks, quantum Markov chains and recurrence

In the present paper, we construct QMCs (Quantum Markov Chains) associated with Open Quantum Random Walks such that the transition operator of the chain is defined by OQRW and the restriction of QMC to the commutative subalgebra coincides with the distribution [Formula: see text] of OQRW. This sheds new light on some properties of the measure [Formula: see text]. As an example, we simply mention that the measure can be considered as a distribution of some functions of certain Markov processes. Furthermore, we study several properties of QMC and associated measures. A new notion of [Formula: see text]-recurrence of QMC is studied, and the relations between the concepts of recurrence introduced in this paper and the existing ones are established.

Mean hitting times of quantum Markov chains in terms of generalized inverses

We study quantum Markov chains on graphs, described by completely positive maps, following the model due to S. Gudder (J. Math. Phys. 49, 072105, 2008) and which includes the dynamics given by open quantum random walks as defined by S. Attal et al. (J. Stat. Phys. 147:832-852, 2012). After reviewing such structures we examine a quantum notion of mean time of first visit to a chosen vertex. However, instead of making direct use of the definition as it is usually done, we focus on expressions for such quantity in terms of generalized inverses associated with the walk and most particularly the so-called fundamental matrix. Such objects are in close analogy with the theory of Markov chains and the methods described here allow us to calculate examples that illustrate similarities and differences between the quantum and classical settings.

Quantum Markov Chains Associated with Open Quantum Random Walks

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

Quantum Dynamical Mutual Entropy Based on AOW Entropy

The classical dynamical mutual entropy measures the average of information content going through a channel. Classical Markovian sources are important in communication theory since they constitute reasonable models for languages. In this paper, we define the quantum dynamical mutual entropy through quantum Markov chains, and we calculate it for several simple models.

Open Quantum Random Walks and Quantum Markov Chains

In the present paper we construct quantum Markov chains associated with open quantum random walks in the sense that the transition operator of a chain is determined by an open quantum random walk and the restriction of the chain to the commutative subalgebra coincides with the distribution ℙρ of the walk. This sheds new light on some properties of the measure ℙρ. For example, this measure can be considered as the distribution of some functions of a certain Markov process.

Clustering Property of Quantum Markov Chain Associated to XY-model with Competing Ising Interactions on the Cayley Tree of Order Two

In the present paper we consider a Quantum Markov Chain (QMC) corresponding to the XY -model with competing Ising interactions on the Cayley tree of order two. Earlier, using finite volumes states one has been constructed QMC as a weak limit of those states which depends on the boundary conditions. It was proved that the limit state does exist and not depend on the boundary conditions, i.e. it is unique. In the present paper, we establish that the unique QMC has the clustering property, i.e. it is mixing with respect to translations of the tree. This means that the von Neumann algebra generated by this state is a factor.

Open Quantum Random Walks and Quantum Markov Chains

В работе строятся КМЦ (квантовые марковские цепи), связанные с открытыми квантовыми случайными блужданиями, так, что переходный оператор цепи задается блужданием, а ограничение КМЦ на коммутативную подалгебру совпадает с распределением $\mathbb{P}_\rho$ блуждания. Это проливает новый свет на некоторые свойства меры $\mathbb{P}_\rho$. В качестве примера мы отмечаем, что эту меру можно рассматривать как распределение некоторых функций некоторого марковского процесса.

Quantum Markov processes: From attractor structure to explicit forms of asymptotic states

Markov processes play an important role in physics and the theory of open systems in particular. In this paper we study the asymptotic evolution of trace-nonincreasing homogenous quantum Markov processes (both types, discrete quantum Markov chains and continues quantum dynamical semigroups) equipped with a so-called strictly positive T -state in the Schrodinger and the Heisenberg picture. We derive a fundamental theorem specifying the structure of the asymptotic and uncover a rich set of transformations between attractors of quantum Markov processes in both pictures. Moreover, we generalize the structure theorem derived for quantum Markov chains to quantum dynamical semigroups showing how the internal structure of generators of quantum Markov processes determines attractors in both pictures. Based on these results we provide two characterizations of all asymptotic and stationary states, both strongly reminding in form the well-known Gibbs states of statistical mechanics. We prove that the dynamics within the asymptotic space is of unitary type, i.e. quantum Markov processes preserve a certain scalar product of operators from the asymptotic space, but there is no corresponding unitary evolution on the original Hilbert space of pure states. Finally simple examples illustrating the derived theory are given.

Perfect sampling for quantum Gibbs states

We show how to obtain perfect samples from a quantum Gibbs state on a quantum computer. To do so, we adapt one of the ``Coupling from the Past''-algorithms proposed by Propp and Wilson. The algorithm has a probabilistic run-time and produces perfect samples without any previous knowledge of the mixing time of a quantum Markov chain. To implement it, we assume we are able to perform the phase estimation algorithm for the underlying Hamiltonian and implement a quantum Markov chain such that the transition probabilities between eigenstates only depend on their energy. We provide some examples of quantum Markov chains that satisfy these conditions and analyze the expected run-time of the algorithm, which depends strongly on the degeneracy of the underlying Hamiltonian. For Hamiltonians with highly degenerate spectrum, it is efficient, as it is polylogarithmic in the dimension and linear in the mixing time. For non-degenerate spectra, its runtime is essentially the same as its classical counterpart, which is linear in the mixing time and quadratic in the dimension, up to a logarithmic factor in the dimension. We analyze the circuit depth necessary to implement it, which is proportional to the sum of the depth necessary to implement one step of the quantum Markov chain and one phase estimation. This algorithm is stable under noise in the implementation of different steps. We also briefly discuss how to adapt different ``Coupling from the Past''-algorithms to the quantum setting.

Decomposition of quantum Markov chains and its applications

Markov chains have been widely employed as a fundamental model in the studies of probabilistic and stochastic communicating and concurrent systems. It is well-understood that decomposition techniques play a key role in reachability analysis and model-checking of Markov chains. (Discrete-time) quantum Markov chains have been introduced as a model of quantum communicating systems [1] and also a semantic model of quantum programs [2]. The BSCC (Bottom Strongly Connected Component) and stationary coherence decompositions of quantum Markov chains were introduced in [3–5]. This paper presents a new decomposition technique, namely periodic decomposition, for quantum Markov chains. We further establish a limit theorem for them. As an application, an algorithm to find a maximum dimensional noiseless subsystem of a quantum communicating system is given using decomposition techniques of quantum Markov chains.

Uniqueness of Quantum Markov Chain Associated with XY-Ising Model on Cayley Tree of Order Two

In this paper, we consider backward and forward Quantum Markov Chains (QMC) associated with XY -Ising model on the Cayley tree of order two. We construct finite volume states with boundary conditions, and define QMC as a weak limit of those states which depend on the boundary conditions. We prove that the limit state is a unique QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Moreover, we observe the relation between backward and forward QMC.

On Translation Invariant Quantum Markov Chains associated with Ising-XY models on a Cayley tree

In this paper, we consider the Ising-XY model with competing interactions on the Cayley tree of order two. This model can be seen as a non-commutative (i.e. J-XY -interactions on next-neighbor vertices) perturbation of the classical Ising model on the Cayley tree. For the considered model we establish the existence of three translation-invariant quantum Markov chains. We notice that if the XY -interactions vanish, i.e. J = 0, then one gets the Ising model. If the classical Ising model vanishes in the considered model, then we obtain XY -model for which it turns out there exists only one translation invariant QMC.

Recurrence and Transience within Quantum Markov Chains

We introduce a new notion of recurrence and transience for the Quantum Markov Chains which are weaker than Accardi-Koroliuk’s recurrence and transience, respectively. We investigate some its properties.