A Quantum Dynamical Approach to Matrix Khrushchev's Formulas

Abstract

Abstract. Khrushchev's formula is the cornerstone of the so called Khrushchev theory, a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle. This formula can be understood as a factorization of the Schur function for an orthogonal polynomial modification of a measure on the unit circle. No such formula is known in the case of matrix-valued measures. This constitutes the main obstacle to generalize Khrushchev theory to the matrix-valued setting which we overcome in this paper.It was recently discovered that orthogonal polynomials on the unit circle and their matrix-valued versions play a significant role in the study of quantum walks, the quantum mechanical analogue of random walks. In particular, Schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system, a topic in which Khrushchev's formula has profound and surprising implications.We will show that this connection between Schur functions and quantum walks is behind a simple proof of Khrushchev's formula via 'quantum' diagrammatic techniques for CMV matrices. This does not merely give a quantum meaning to a known mathematical result, since the diagrammatic proof also works for matrix-valued measures. Actually, this path counting approach is so fruitful that it provides different matrix generalizations of Khrushchev's formula, some of them new even in the case of scalar measures.Furthermore, the path counting approach allows us to identify the properties of CMV matrices which are responsible for Khrushchev's formula. On the one hand, this helps to formalize and unify the diagrammatic proofs using simple operator theory tools. On the other hand, this is the origin of our main result which extends Khrushchev's formula beyond the CMV case, as a factorization rule for Schur functions related to general unitary operators.

Complete homotopy invariants for translation invariant symmetric quantum walks on a chain

“…Similar to the case of symmetry type D, the additional invariant can be trivialized, by either regrouping neighbouring cells once or by adding trivial systems to the respective walks under consideration: Proof. If we regroup a given walk once, according to (13), we get c r (k) = c(k/2)c(k/2 + π) and hence c r (0) = c(0)c(π) = c(0) 2 (−1) p = (−1) p , which only depends on the winding number and is therefore the same for W 1 and W 2 . Now let W 0 i = Z W i (0) be the walk, which is block diagonal, with W i (0) acting locally in each cell.…”

Singular continuous Cantor spectrum for magnetic quantum walks

“…At the same time quantum walks and their time-continuous counterpart are relevant in a quantum algorithmical context, for example in search algorithms, to examine the distinctness of elements, in quantum information processing and applications to the graph isomorphism problem [6,11,27,28,48,51]. From a mathematical point of view, quantum walks on a one-dimensional lattice can also be seen as a particular class of CMV matrices giving a link between quantum dynamical systems and orthogonal polynomials on the unit circle, a connection which has proved to be fruitful in both directions [12,15,16,19,24,39]. One can use spectral methods to deduce bounds on spreading [31,32].…”

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

“…One of the most interesting topics in the theory of orthogonal polynomials on the unit circle (OPUC) (see among many classic references) is its connections with other areas of current interest, like Khrushchev theory, quantum walks, Cantero, Moral and Velázquez (CMV) matrices, and asymptotic quantum algorithm for the Toeplitz systems (see, for example, Cedzich et al and Lin‐Chun and references therein).…”