Carlos F. Lardizabal

On a Class of Quantum Channels, Open Random Walks and Recurrence

We study a particular class of trace-preserving completely positive maps, called PQ-channels, for which classical and quantum evolutions are isolated in a certain sense. By combining open quantum random walks with a notion of recurrence, we are able to describe criteria for recurrence of the walk related to this class of channels. Positive recurrence for open walks is also discussed in this context.

Open Quantum Random Walks: Ergodicity, Hitting Times, Gambler’s Ruin and Potential Theory

In this work we study certain aspects of open quantum random walks (OQRWs), a class of quantum channels described by Attal et al. (J Stat Phys 147: 832–852, 2012). As a first objective we consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by Saloff-Coste and Zúñiga (Stoch Proc Appl 117: 961–979, 2007), we define a notion of ergodicity for finite nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a second objective, and based on a quantum trajectory approach, we study a notion of hitting time for OQRWs and we see that many constructions are variations of well-known classical probability results, with the density matrix degree of freedom on each site giving rise to systems which are seen to be nonclassical. In this way we are able to examine open quantum versions of the gambler’s ruin, birth-and-death chain and a basic theorem on potential theory.

A Thermodynamic Formalism for Density Matrices in Quantum Information

We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the denitions and results we describe here are natural generalizations of the classical concepts of Thermo- dynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is well-known that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism. We consider the operatoracting on the space of density matr ices MN over anite N-dimensional complex Hilbert space �( �) := k X i=1 tr(WiW � i ) ViV � i tr(ViV � i ) ,

Site recurrence of open and unitary quantum walks on the line

We study the problem of site recurrence of discrete-time nearest-neighbor open quantum random walks (OQWs) on the integer line, proving basic properties and some of its relations with the corresponding problem for unitary (coined) quantum walks (UQWs). For both kinds of walks, our discussion concerns two notions of recurrence, one given by a monitoring procedure (Grünbaum et al. in Commun Math Phys 320:543–569, 2013; Lardizabal and Souza in J Stat Phys 159:772–796, 2015), and we study their similarities and differences. In particular, by considering UQWs and OQWs induced by the same pair of matrices, we discuss the fact that recurrence of these walks is related by an additive interference term in a simple way. Based on a previous result of positive recurrence, we describe an open quantum version of Kac’s lemma for the expected return time to a site.

A quantization procedure based on completely positive maps and Markov operators

We describe ω-limit sets of completely positive (CP) maps over finite-dimensional spaces. In such sets and in its corresponding convex hulls, CP maps present isometric behavior and the states contained in it commute with each other. Motivated by these facts, we describe a quantization procedure based on CP maps which are induced by Markov (transfer) operators. Classical dynamics are described by an action over essentially bounded functions. A non-expansive linear map, which depends on a choice of a probability measure, is the centerpiece connecting phenomena over function and matrix spaces.

A Dynamical Point of View of Quantum Information: Discrete Wigner Measures

We describe some well known properties of Wigner measures and then analyze some connections with Quantum Iterated Function Systems.

Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster’s Theorem

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin–McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler’s ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster’s Theorem for the expected return time together with applications.

A dynamical point of view of Quantum Information: Wigner measures

We analyze a known version of the discrete Wigner function and some connections with Quantum Iterated Funcion Systems. Dynamics, Games and Science II, DYNA 2008, Edit. M. Peixoto, A. Pinto and D Rand, Springer Verlag (2011) 1 Discrete Weyl relations This section follows parts of [3]. Consider the Hilbert space H = C . Let {|k〉} k=0 be an orthonormal base. Fix αu, αv ∈ [0, 1] and define the following matrices UN , VN ∈MN (C): UN := e 2π N iαu N−1 ∑ k=0 e 2π N |k〉〈k|, VN := e 2π N iαv N−1 ∑ k=0 |k〉〈k − 1| (1) together with the identification |j〉 = |j mod N〉. Such operators are unitary and we have A. T. Baraviera I.M. UFRGS, Porto Alegre 91500-000, Brasil, e-mail: atbaraviera@gmail.com C. F. Lardizabal I.M. UFRGS, Porto Alegre 91500-000, Brasil, e-mail: carlos.lardizabal@gmail.com A. O. Lopes I.M. UFRGS, Porto Alegre 91500-000, Brasil, e-mail: arturoscar.lopes@gmail.com M. Terra Cunha D. M UFMG, Belo Horizonte 30161-970, Brasil, e-mail: marcelo.terra.cunha@gmail.com

A Dynamical Point of View of Quantum Information: Entropy and Pressure

Quantum Information is a new area of research which has been growing rapidly since last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more “dynamical point of view” of this theory. We want to consider the concepts of entropy and pressure for “stationary systems” acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the ThermodynamicFormalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator ℒ acting on density matrices ρ ∈ ℳ N over a finite N-dimensional complex Hilbert space ℒ(ρ) : = ∑ i = 1 k tr(W i ρW i ∗ )V i ρV i ∗ , where W i and V i , i = 1, 2, …k are operators in this Hilbert space. ℒ is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the V i (. )V i ∗ = : F i (. ), i = 1, 2, …, k, play the role of the inverse branches (acting on the configuration space of density matrices ρ) and the W i play the role of the weights one can consider on the IFS. We suppose that for all ρ we13pc]First author considered as corresponding author. Please check. have that ∑ i = 1 k { tr}(W i ρW i ∗ ) = 1. A family W : = { W i } i = 1, …, k determines a Quantum Iterated Function System (QIFS) ℱ W , ℱ W = { ℳ N , F i , W i } i = 1, …, k .