Quantum Recurrence of a Subspace and Operator-Valued Schur Functions

A notion of monitored recurrence for discrete-time quantum processes was recently introduced in Grunbaum et al. (Commun Math Phys (2), 320:543-569, 2013) taking the initial state as an absorbing one. We extend this notion of monitored recurrence to absorbing subspaces of arbitrary finite dimension. The generating function approach leads to a connection with the well-known theory of operator-valued Schur functions. This is the cornerstone of a spectral characterization of subspace recurrence that generalizes some of the main results in Grunbaum et al. (Commun Math Phys (2), 320:543-569, 2013). The spectral decomposition of the unitary step operator driving the evolution yields a spectral measure, which we project onto the subspace to obtain a new spectral measure that is purely singular iff the subspace is recurrent, and consists of a pure point spectrum with a finite number of masses precisely when all states in the subspace have a finite expected return time. This notion of subspace recurrence also links the concept of expected return time to an Aharonov-Anandan phase that, in contrast to the case of state recurrence, can be non-integer. Even more surprising is the fact that averaging such geometrical phases over the absorbing subspace yields an integer with a topological meaning, so that the averaged expected return time is always a rational number. Moreover, state recurrence can occasionally give higher return probabilities than subspace recurrence, a fact that reveals once more the counter-intuitive behavior of quantum systems. All these phenomena are illustrated with explicit examples, including as a natural application the analysis of site recurrence for coined walks.