1202.3903 (F. A. Grünbaum et al.)

Recurrence for discrete time unitary evolutions    [PDF]

F. A. Grünbaum, L. Velázquez, A. H. Werner, R. F. Werner

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. We also
show that in the recurrent case the expected first return time is an integer or
infinite, for which we give a topological interpretation. A key role in our
theory is played by the first arrival amplitudes, which turn out to be the
(complex conjugated) Taylor coefficients of the Schur function of the spectral
measure. On the one hand, this provides a direct dynamical interpretation of
these coefficients; on the other hand it links our definition of first return
times to a large body of mathematical literature.

View original: http://arxiv.org/abs/1202.3903