  What happens when a quantum system undergoing unitary evolution in time is
subject to repeated projective measurements to the initial state at random
times? A question of general interest is: How does the survival probability
$S_m$, namely, the probability that an initial state survives even after $m$
number of measurements, behave as a function of $m$? We address these issues in
the context of two paradigmatic quantum systems, one, the quantum random walk
evolving in discrete time, and the other, the tight-binding model evolving in
continuous time, with both defined on a one-dimensional periodic lattice with a
finite number of sites $N$. For these two models, we present several numerical
and analytical results that hint at the curious nature of quantum measurement
dynamics. In particular, we unveil that when evolution after every projective
measurement continues with the projected component of the instantaneous state,
the average and the typical survival probability decay as an exponential in $m$
for large $m$. By contrast, if the evolution continues with the leftover
component, namely, what remains of the instantaneous state after a measurement
has been performed, the survival probability exhibits two characteristic $m$
values, namely, $m_1^\star(N) \sim N$ and $m_2^\star(N) \sim N^\delta$ with
$\delta >1$. These scales are such that (i) for $m$ large and satisfying $m <
m_1^\star(N)$, the decay of the survival probability is as $m^{-2}$, (ii) for
$m$ satisfying $m_1^\star(N) \ll m < m_2^\star(N)$, the decay is as $m^{-3/2}$,
while (iii) for $m \gg m_2^\star(N)$, the decay is as an exponential. We find
that our results hold independently of the choice of the distribution of times
between successive measurements, as have been corroborated by our results for a
wide range of distributions. This fact hints at robustness and ubiquity of our
derived results.
