Recurrence for discrete time unitary evolutions

Understand

We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \phi.

• In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state.
• We call the system recurrent if this eventually happens with probability one.
• We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \phi.