Quantum recurrence of a subspace and operator-valued Schur functions (Q2249776)

This article explores subspace recurrence for quantum dynamical systems. Recall that a quantum dynamical system consists of a Hilbert space \(\mathcal H\) and a unitary operator \(U\) on \(\mathcal H\); let now \(V\) be a closed subspace and let \(P\) denote orthogonal projection to \(V\). The quantum random process is then defined on functions as follows. Given an initial state \(\psi\in V\) of norm \(1\), repeatedly apply \(U\) followed by the projection to \(V^\perp\) followed by normalization, unless the vector belongs to \(V\). Therefore, the random process's evolution is a scalar multiple of \(\tilde U:=(1-P)U\). Following Born's probabilistic interpretation, \(\|\tilde U^n\psi\|^2\) is the probability that the process did not yet return to \(V\) in \(n\) steps, and \(\|PU\tilde U^{n-1}P\psi\|^2\) is the \(n\)-step first \(V\)-return probability of \(\psi\). The state \(\psi\) is called ``\(V\)-recurrent'' if these probabilities sum to \(1\). Note that the random process interacts with \(V\) at each step, by projection. The probabilities \(\|PU^n\psi\|^2\) can therefore be dramatically different, and the authors explain in some detail an example in which \(V\) is not recurrent, but some subspaces are recurrent. The reader will find a wealth of examples, illustrations and explanations in this article, but few theorems. The main one gives a characterization of subspaces \(V\) all of whose states are \(V\)-recurrent with finite expected \(V\)-return time. The main tool used are operator-valued generating functions.